Ricci Flows of Ricci flat Cones

Ricci Flows of Ricci flat Cones - ETH E-Collection

DISS. ETH No. 21252
Ricci Flows of Ricci flat Cones
A dissertation submitted to
Eth
Zurich
for the degree of
DOCTOR
OF
SCIENCES
presented by
Michael
Siepmann
Dipl. Math. ETH, ETH Zürich
Date of birth
December 6th, 1981
citizen of
Germany
accepted on the recommendation of
Prof. Dr. Tom Ilmanen, examiner
Prof. Dr. Lei Ni, co-examiner
Prof. Dr. Michael Struwe, co-examiner
2013
Abstract
In this thesis we study the evolution of smooth Riemannian cones under Hamilton’s Ricci flow. In the first
part we define Ricci flows coming out of a Riemannian cone and prove using Perelman’s Pseudolocality
Theorem that the quadratic curvature decay is preserved. In fact, in combination with Shi’s estimates
all derivatives decay at the appropriate rate for t > 0 as x → ∞. We go on to consider Ricci flat cones as
initial conditions. By construction of a suitable supersolution we show that the Ricci curvature decays
exponentially for any smooth Ricci flow coming out of a Ricci flat cone. This result corresponds to the
exponential curvature decay for the 2-dimensional expanding gradient Ricci solitons coming out of Ricci
flat cones described in [CLN06] and the Kähler gradient Ricci solitons in [FIK03]. Inspiration comes
from similar results for mean curvature flows from stationary cones in [Ilm95].
In the second part we prove that gradient expanding Ricci solitons with quadratic curvature decay are
solutions of the Ricci flow coming out of a cone as defined in the first part. To this end we show that
the gradient potential f converges as t & 0 to a continuous function on M and locally smoothly outside
of a compact subset. The deRham splitting theorem then gives the initial cone.
The third part also contains the main result of this thesis. We construct expanding Kähler Ricci solitons
coming out of Ricci flat Kähler cones. We show that if a Ricci flat Kähler cone (a Riemannian cone
over a Sasaki Einstein manifold) admits an equivariant resolution with positive canonical line bundle
(negative first Chern class) then there exists an expanding self-similar solution coming out of that cone
as defined in the first part. By the work of van Coevering Kähler cones can always be understood as
complex varieties and we follow the arguments there to set up the corresponding complex Monge Ampere
equation on such a resolution. This complex Monge Ampere equation is elliptic and has unbounded
coefficients in the first order term. To show existence of a solution we combine Aubin’s and Yau’s
estimates for the proof the the Calabi conjecture with technique’s developed by Lunardi to deal with
elliptic operators with unbounded coefficients acting on certain weighted spaces. Finally we present some
old examples of expanding Kähler Ricci solitons in the light of this of this theorem and also construct
new examples using toric geometry.
We conclude this thesis with a list of open problems and possible directions for further research.
i
Zusammenfassung
In der vorliegenden Arbeit untersuchen wir das Verhalten des Ricci-Flusses glatter Riemann’scher Kegel.
Im ersten Teil der Arbeit geben wir eine geeignete Definition für Ricci Flüsse aus Riemann’schen Kegel
und zeigen mit Hilfe von Perelman’s Pseudolokalitätstheorem, dass der quadratische Krümmungsabfall
unter dem Ricci-Fluss erhalten bleibt. In Kombination mit Shis Krümmungsabschätzungen weisen wir
nach, dass auch die höheren kovarianten Ableitungen der Krümmung mit entsprechenden Raten abfallen.
Anschliessend erforschen wir Ricci Flüsse welche Ricci-flachen Kegeln entspringen. Durch Konstruktion
einer geeigneten Super-Lösung können wir zeigen, dass die Ricci-Krümmung für positive Zeiten beliebiger Ricci Flüsse aus Ricci-flachen Kegeln exponentiell abfällt. Man vergleiche dieses Resultat mit
entsprechenden Beobachtungen exponentiellen Krümmungsabfalls für die 2-dimensionalen expandierenden selbstähnlichen Lösungen des Ricci-Flusses aus Ricci-flachen Kegeln in [CLN06] sowie denjenigen
Lösungen, welche in [FIK03] konstruiert wurden. Die Inspiration hierfür entstammt unter anderem
ähnlichen Resultaten für den mittleren Krümmungsfluss aus stationären Kegeln.
Im zweiten Teil weisen wir nach, dass expandierende Gradienten Ricci Solitone mit quadratischem
Krümmungsabfall Lösungen des Ricci-Flusses aus Riemann’schen Kegeln Kegeln im Sinne des ersten
Teils sind. Dazu zeigen wir, dass das Gradientenpotential des gegebenen Ricci Solitons gegen ein stetige
Funktion und lokal ausserhalb einer kompakten Teilmenge gegen eine glatte Funktion konvergiert, wenn
t gegen 0 strebt. Der Spaltungssatz von de Rham impliziert sodann das angekündigte Resultat.
Der dritte Teil stellt auch den Hauptteil dieser Arbeit dar. Wir konstruieren Kähler Ricci Solitone,
welche Ricci flachen Kähler Kegeln (Riemannschen Kegeln über Sasaki-Einstein Mannigfaltigkeiten)
entspringen. Es wird gezeigt, dass Ricci-flache Kegel, welche eine equivariante Auflösung mit positivem
kanonischen Linienbündel (negativer erster Chern Klasse) zulassen, immer selbstähnliche Lösungen des
Kähler Ricci Flusses aus diesen Kegeln im obigen Sinnes besitzen. Gemäss der Arbeit von van Coevering können Kähler Kegel stets als komplexe Varietäten verstanden werden, und wir folgen den dort
entwickelten Argumenten, um die entsprechenden Monge Ampere Gleichungen für die Existenz dieser
Lösungen aufzustellen. Der Nachweis der Existenz von Lösungen dieser skalaren komplexen Monge
Ampere Gleichung verwendet die Abschätzungen von Aubin und Yau für den Beweis der Calabi Vermutung sowie Techniken von Lunardi, welche elliptische Operatoren mit unbeschränkten Operatoren auf
gewissen gewichteten Funtionenräumen betreffen. Anschliessend präsentieren wir einige alte Beispiele
expandierender Kähler Ricci Solitone im Lichte dieses Satzes und konstruieren auch neue Beispiele aus
dem Bereich der torischen Geometrie.
Wir schliessen mit einer Reihe offener Fragen sowie möglichen zukünftigen Richtungen unserer Forschung.
iii
Contents
Abstract
i
Zusammenfassung
iii
Chapter 1. Introduction
1.1. Ricci flow through singularities
1.2. Comparison with other flows
1.3. Non-uniqueness for Ricci flow and Ricci flow of cones
1.4. Reverse bubbling
1.5. Kähler Ricci flow
1.6. Asymptotically conical Ricci flat Kähler manifolds
1.7. Overview
1
1
2
2
4
5
5
6
Chapter 2. Preliminaries
2.1. Kähler Geometry
2.2. Notions of Positivity
2.3. Monge Ampere Equations and the Calabi Yau theorem
2.4. Kähler Ricci flow
2.5. Sasakian Geometry
2.6. Sasaki Einstein manifolds
2.7. Kähler Cones
2.8. Ricci flat Kähler cones
2.9. Asymptotically conical manifolds
9
9
14
14
16
17
19
20
21
23
Chapter 3. Ricci Flows attaining the Cone as an initial condition
3.1. Introduction
3.2. Ricci flows coming out of cones
3.3. Ricci flows coming out of Ricci flat cones
25
25
26
28
Chapter 4. Expanding Ricci Solitons
4.1. Introduction
4.2. Basic properties of expanding Ricci solitons
4.3. Convergence of Ricci solitons
35
35
35
37
Chapter 5. Existence of Expanding Kähler Ricci Solitons
5.1. Introduction
5.2. Kähler Ricci solitons
5.3. Statement of the Existence Theorem
5.4. Proof of the existence theorem
5.5. Old Examples of Expanding Kähler Ricci Solitons
5.6. New Examples of Expanding Kähler Ricci Solitons
41
41
41
43
46
69
70
Chapter 6.
75
Open Problems
Appendix A. Schauder Estimates for Ornstein Uhlenbeck Operators
A.1. Global Schauder estimates on Rn
v
79
79
vi
CONTENTS
A.2.
A.3.
Set up for M with a conical end
Global Schauder estimates on M
79
80
Appendix B. Kähler cones as complex varieties
B.1. General Kähler cones
B.2. Ricci flat Kähler Cones
83
83
86
Appendix C. Toric Manifolds and Toric resolutions
C.1. Symplectic Toric Manifolds
C.2. Toric Varieties
C.3. Toric Kähler geometry
89
89
91
94
Appendix D. Maximum Principles and Schauder Estimates
D.1. Maximum Principles
D.2. Schauder Estimates
99
99
100
Bibliography
101
Acknowledgements
107
CHAPTER 1
Introduction
Geometric heat flows have a long and successful history. Presumably starting with the harmonic map
heat flow introduced by Eels and Sampson [ES64] these geometric flows have helped to solve various
problems in geometry and topology. One of the more famous exponents is the Ricci flow introduced
by Hamilton in his seminal work [Ham82] from 1982 which deforms a Riemannian metric g by the
evolution equation
∂gij
= −2Rcij (g(t))
∂t
towards a canonical metric. In this paper and subsequent work Hamilton laid out a program using Ricci
flow to approach Thurston’s geometrization conjecture which entails the famous Poincaré conjecture as
a corollary. In the following decades Hamilton and numerous others contributed to the execution of
this program culminating in the celebrated work of Perelman ([Per02], [Per03b], and [Per03a]) whose
results completed the proof of Thurston’s conjecture (see [MT07], [KL08]). While the Ricci flow was
originally developed only for 3-dimensional manifolds it has also been successfully applied in higher
dimensions for example in the work of Böhm and Wilking [BW08] who proved, using Ricci flow, that
manifolds with positive curvature are space forms and more recently in Brendle and Schoen’s proof of
the differentiable sphere conjecture [BS11].
Parallel to this development Ricci flow opened up a new field of research in Kähler geometry (with its
own geometrization program) that has grown rapidly in recent years. The Ricci flow equation on Kähler
manifolds reduces to a scalar Monge Ampere equation and even in higher dimensions the formation of
singularities is much better understood, at least in the case of algebraic varieties.
Here we are interested in the “un-formation” of singularities under Ricci flow in general and Kähler
Ricci flow in particular. As a heat flow the Ricci flow has the tendency to improve regularity and in
this thesis we investigate whether the Ricci flow can smoothen out certain initial singularities. Apart
from that we also study possible non-uniqueness issues after the formation of singularities.
Our driving motivation is a question raised in [FIK03] which contains both of these aspects: When does
a Ricci flat Kähler cone (a stationary cone for the Ricci flow) posses a non-singular forward evolution
by Ricci flow? This is illustrated by the following example in [FIK03]: the orbifold Cn /Zk can be
equipped with the flat metric g0 to become a Ricci flat cone. At the same time, for k > n, it is shown
that there exists a smooth Ricci flow (M, g(t))t>0 which converges to the Ricci flat cone as t & 0 in the
Gromov Hausdorff sense and locally smoothly away from the vertex. These Ricci flows generally do not
have curvature bounded from below. The goal of this thesis is to further generalize this example and
provide a general necessary criterion for the existence of such flows.
In the following sections we present some related topics and results that guide and motivate our research.
In the last section we give an overview of the chapters that follow and the main results of this thesis.
1.1. Ricci flow through singularities
In general the Ricci flow develops singularities. To continue the flow past such singularities Perelman
introduced a surgery algorithm. The basic idea is to replace the parts of the manifold where the
curvature blows up just before the singular time by a smooth piece in such a way that it satisfies
certain a priori estimates and restart the flow there. However, this procedure depends on a number of
choices. So Perelman suggests that there should be a way of flowing “into” the singularity and out of
the resulting singular space in an unambiguous way. Such a procedure is described in [ACK09]. This
is one motivation to study Ricci flows with singular initial conditions.
1
2
1. INTRODUCTION
1.2. Comparison with other flows
It is instructive to compare the behavior of the Ricci flow near and at singularities with similar situations
for other geometric flows. For the three flows described below non-uniqueness phenomena for stationary
cones have been described in the literature (see [Ilm95] and references therein).
1.2.1. Harmonic map heat flow. For the harmonic map heat flow (in supercritical dimension
n ≥ 3) an example of a stationary cone is the map u0 : Rn → S n , x 7→ (π/2, x/|x|) where (s, ω) are
corotational coordinates on S n . The map u0 is weakly harmonic and hence in particular a self-similar
(weak) solution of the harmonic map heat flow. In [Ilm95] (see also [GR11] for a more general result) it
is shown that there exists a non-constant solution of the harmonic map heat flow with initial condition u0
if u0 is not energy minimizing. In particular, Angenent, Ilmanen, and Velazquez [Ilm95] (see [GR11])
show that there exists a self similar solution not identical with the cone if 3 ≤ n < 7. Other examples
of non-uniqueness were found by Coron [Cor90]. In critical dimension 2 the weak flow is unique under
the additional requirement that the energy is monotone ([Str85], [Fre95]).
1.2.2. Mean curvature flow. For the mean curvature flow the stationary cones are submanifolds
which are invariant under homotheties and have vanishing mean curvature (on the smooth part, i.e.
away from the origin). It is shown in [Ilm95] that a codimension one stationary cone has a non-unique
evolution if it is not area minimizing. As an example consider the Simons cones which are defined, for
p + q ≥ 2, as
Cp,q := {(x, y) ∈ Rp × Rq : |y|2 /(p − 1) = |x|2 /(q − 1)}.
Simons showed in [Sim68] that for p + q > 7 these cones are area minimizing and Bombieri, de Giorgi,
and Giusti showed in [BDGG69] that these cones are unstable (i.e. not area minimizing) if p + q ≤ 7.
This reflects a general result that the singular set of an area minimizing hypersurface has codimension
7. In the cases p + q ≤ 7 there exists a self similarly expanding solution different from Cp,q (see [Ilm95]
and example 3.3.5 in this thesis).
1.2.3. Lagrangian mean curvature flow. Lagrangian mean curvature flow is a special case of
mean curvature flow. If M is a Lagrangian submanifold of a Calabi Yau manifold then the Lagrangian
property is preserved under the mean curvature flow. A particularly simple example of a stationary Lagrangian cone is the union of two Lagrangian n-planes in Cn which intersect only in the origin. Lawlor
and Nance ([Law89] and [Mac87]) showed that such a cone is area minimizing if and only if it satisfies
an angle criterion. Joyce, Lee and Tsui [JLT10] constructed self-similar solutions coming out of such
cones that do not satisfy the angle criterion. Moreover, it is shown in [LN12] that these are the only
Lagrangian expanders with zero Maslov class for each pair of 2-planes in C2 .
1.2.4. Mean curvature flow and Ricci flow. Mean curvature flow and Ricci flow bear several
interesting similarities. In particular, mean curvature flow in dimension 2 and Ricci flow in dimension
4 seem to be closely related. However, this analogy does not work if one only considers mean curvature
flows in codimension 1 (see [HHS11]). There is much less known about mean curvature flow and
minimal submanifolds in higher codimension (e.g. minimal surfaces in higher codimension have only
isolated singular points). It is expected that some results for mean curvature flow of Lagrangian surfaces
in C2 (which is much better understood) will have their analogue in the Kähler Ricci flow on complex
surfaces and vice versa. But even this analogy breaks down in general (see [Nev07], [FIK03]).
In this spirit the singular set of a Ricci flow should have (parabolic) codimension 4 which corresponds
to Almgren’s regularity result [Alm00] for area minimizing currents in general codimension (see also
[Whi00]). In particular, this would make Ricci flat cones valid solutions of the Ricci flow (see remarks
in [FIK03]).
1.3. Non-uniqueness for Ricci flow and Ricci flow of cones
On a smooth manifold a solution of the Ricci flow is unique if the curvature is bounded along the flow
and the initial metric is complete ([CZ06]). Topping showed that uniqueness fails in general if either of
1.3. NON-UNIQUENESS FOR RICCI FLOW AND RICCI FLOW OF CONES
3
these conditions is dropped (see [Top11] and [GT01]). Non-uniqueness also arises for singular initial
conditions. In the following subsections we describe some partial results related to non-uniqueness for
the Ricci flow of Ricci flat cones.
1.3.1. Ricci flat cones. Ricci flat cones are the stationary cones for the Ricci flow. As such they
should be considered solutions of the Ricci flow equation (see remarks above). A special class of Ricci
flat cones are Ricci flat Kähler cones, i.e. cones over Sasaki Einstein manifolds, which are an interesting
topic of research in its own right. These cones have come up in the work of Cheeger, Colding and Tian
[CT11] (note, however, that not every Ricci flat Kähler cone is the limit of Calabi Yau manifolds) and
numerous examples of Sasaki Einstein manifolds have been constructed more recently (see section 2.5).
1.3.2. Stability of Ricci flat cones. As described above, unstable (with respect to some energy
functional) stationary cones cause non-uniqueness for various geometric flows. In [HHS11] we discuss
the relation between unstable cones and non-uniqueness of the Ricci flow. There, a Ricci flat cone is
considered to be stable if the renormalized Perelman functional λnc introduced in [Has11] has a local
maximum in (C, gC ). This means that for any nearby manifold (M, g) asymptotic to the original cone
(C, gC ) we have λnc (g) ≤ λnc (gC ). It is conjectured that in the case that (C, gC ) is not stable a Ricci
flow not identical with the cone exists. However, this flow might very well be singular. Conversely, if a
smooth flow coming out of the Ricci flat cone exists then there exists a competitor (M, g) asymptotic
to the cone with λnc (g) > λnc (gC ) (see [HHS11]).
1.3.3. Ricci flow and the positive mass theorem. It was conjectured by Ilmanen [Ilm] that
there exists a relation between non-uniqueness of Ricci flows of Ricci flat cones and the failure of the
positive mass theorem. Recall that the positive mass theorem is not true in general for ALE manifolds
(see [Dah97] and [LeB88] and the remarks below). The relationship between the two has been studied
in [HHS11]. There the following theorem is proven:
Theorem 1.3.1. Let M n (n ≥ 3) be a complete manifold with one end, and assume the end is diffeomorphic to S n−1/Γ × R, where Γ is a finite subgroup of SO(n). Then the following are equivalent:
(1) (lambda not a local maximum) There exists a metric g on M that agrees with the flat conical
metric outside a compact set such that λnc (g) > 0.
(2) (failure of positive mass) There exists an asymptotically locally euclidean metric g on M such
that Rg ≥ 0 and m(g) < 0.
Examples where the positive mass theorem fails for ALE manifolds were constructed by LeBrun [LeB88].
He shows that there exists a complete scalar flat metric on L−k → CP 1 for k > 2 such that m(g) < 0.
This is also true for the scalar flat metric on toric resolutions of cyclic singularities of C2 in [CS04],
[Joy95] and [ASD12]. This theorem does not give a sufficient criterion for the existence of nonstationary Ricci flow coming out of the cone. However, if such a flow exists then it is shown in [HHS11]
that (1) and (2) hold. The correct converse statement would have to take into account possible singular
non-stationary solutions of the Ricci flow coming out of the cone. Our results for smooth (Kähler) Ricci
flows on toric resolutions of cyclic singularities further support this conjecture.
1.3.4. Expanding Ricci solitons. Ricci solitons model the behavior of the Ricci flow near a
singularity and they appear as blow up limits of a Ricci flow at a singular time. Ricci solitons are
self-similar solutions of the Ricci flow (in other words solutions which move only under scaling and
diffeomorphisms) and at any fixed time they satisfy the equation
2Rc − LX g + λg = 0,
where X generates the diffeomorphisms and the sign of λ determines the scaling is shrinking (λ < 0) or
expanding (λ > 0). Understanding Ricci solitons was essential to carry out Hamilton’s program. While
shrinking Ricci solitons describe Ricci flows flowing into singularities expanding Ricci solitons are Ricci
flows coming out of singularities. This means that expanding Ricci solitons are the first thing to study
if one wants to understand Ricci flows coming out of cones. The behavior of expanding Ricci solitons as
t & 0 can be compared with shrinking Ricci solitons as t % 0. In both cases the flows converge to a cone
under the additional assumption of quadratic curvature decay. Given a cone one expects that a Ricci
4
1. INTRODUCTION
flow and probably an expanding Ricci soliton asymptotic to the cone exist (possibly with a singularity).
On the other hand shrinking solitons seem to be quite rare and hence it is unlikely that a given cone is
the limit of a shrinking Ricci soliton. This means that it is much easier to construct expanding Ricci
solitons (and much harder if not impossible to prove their uniqueness) than their shrinking counterparts. Note that Ricci flat cones are expanding and shrinking Ricci solitons at the same time (with
a singularity though). Examples of such self-similar expanding solutions (coming out of Riemannian
cones) were constructed in [GK04], [FIK03], [DW11], and [SS10]. The examples in [FIK03] are
rotationally symmetric expanding Kähler Ricci solitons on resolutions of quotients of Cn by diagonal
cyclic subgroups U (n). The non-uniqueness of the Ricci flow occurs as follows: Cao showed in [Cao96]
n
that for any p ≥ 1 there exists a smooth expanding
√ Kähler2pRicci soliton metric gCao,p on C giving
¯
/p. The metric also gives a well-defined
a Kähler Ricci flow with initial condition ωgp := −1∂ ∂|z|
Kähler Ricci soliton metric gCao,p on C2 /Zk giving a singular Kähler Ricci flow (Cn /Zk , gCao,p (t)). At
the same time it is shown in [FIK03] that for k > n there exists a smooth expanding Kähler Ricci soliton
metric gFIK,p on the k-th power of the tautological line bundle L−1 over CP n−1 with associated Kähler
Ricci flows gFIK,p (t) coming out of the cone (Cn , gp ). Hence there exist two different Ricci flows starting
from the same cone. In particular, if p = 1 the initial cone is flat and gCao,1 is the euclidean expander
on the quotient while (L−k , gFIK,1 ) is a non-stationary expanding Kähler Ricci soliton coming out of
a Ricci flat cone. It is worth noting that the expanding Kähler Ricci solitons constructed in [FIK03]
are ALE and coincide (in complex dimension 2) with the manifolds which were used to construct the
counterexamples to the positive mass theorem in [LeB88].
Another example of an expanding Ricci soliton coming out of a Ricci flat cone is the 2-dimensional
expanding Ricci soliton from [GHMS03] described in [CLN06] (see also chapter 3). As gradient Ricci
solitons on surfaces they are rotationally symmetric. Note that also higher dimensional gradient Kähler
Ricci solitons have at least a one-dimensional symmetry.
The expanders in [SS10] are obtained as smooth limits of Ricci flows on non-compact manifolds with
positive sectional curvature. The evolving manifolds are homeomorphic to the initial cone (the asymptotic cone). However, none of these flows (with exception of the Gaussian expander) are coming out of
Ricci flat cones.
Finally, we should mention the examples constructed by Futaki and Wang in [FW11] which extend the
shrinking and expanding soliton construction of [Cao85] and [FIK03] to Ricci flat Kähler cones (compared to Cn or quotients of Cn ) to obtain global solutions of the Ricci flow on complex line bundles over
Kähler Einstein manifolds with positive scalar curvature. Note that at time t = 0 their metric g = g(0)
is a cone metric but not Ricci flat. This generalizes the global solutions in [FIK03] which are shrinking
Kähler Ricci solitons on L−1 → CP n−1 , blow down the zero section at t = 0, and then continue as the
corresponding expanding Ricci solitons on Cn from [Cao96]. The examples of expanders constructed
in [FW11] are not complete in general. While the shrinking Ricci solitons extend smoothly to the zero
section the expanding Ricci solitons remain singular along the zero section (i.e. they are incomplete in
the smooth part).
The Kähler Ricci solitons constructed in [DW11] are of cohomogeneity one and live on complex line
bundles over products of Kähler Einstein manifolds. They are generalizations of the solitons constructed
in [FIK03] and they extend smoothly across the zero section. However, only few of them come out of
Ricci flat cones.
As described above Ricci flat cones are critical points of the renormalized Perelman functional and
λnc (C, gC ) = 0. At the same time Perelman’s shrinker entropy of the cone is ν− (C) = log(Vol(S)) −
log(Vol(S n−1 )) where S is the link of he cone and Vol(S n−1 ) is the volume of the round sphere (see
[CHI04]). The expander entropy ν+ as defined in [FIN05] on the other hand is not finite on Ricci flat
cones in general. In chapter 6 we speculate about the existence of a relative expander entropy.
1.4. Reverse bubbling
The term “reverse bubbling” was originally used by Topping to describe a phenomenon occurring in
harmonic map heat flow. In general this flow develops singularities and does so (in critical dimension
1.6. ASYMPTOTICALLY CONICAL RICCI FLAT KÄHLER MANIFOLDS
5
2) by blowing up the gradient of the evolving map while concentrating energy in one point (or isolated
points). One says that the flow generates a bubble which causes the topology of the map (the image)
to change. To continue the flow one removes all the bubbles and restarts the flow smoothly. During
the process the energy drops by a fixed amount. Struwe [Str85] and Freire [Fre95] showed that there
exists a unique global solution with decreasing energy. Topping subsequently investigated possible nonuniqueness by inserting bubbles at the singular times (therefore increasing the energy) whose energy
deconcentrates for positive times. Rupflin showed in [Rup08] that this is the only way non-uniqueness
can occur. For the Ricci flow on surfaces a similar phenomenon can be observed. Recall that the rescaled
Ricci flow on a compact Riemannian surface exists globally and converges to a constant curvature
metric so no forward singularities are expected. Reverse bubbles, on the other hand, can be created by
either starting with an incomplete metric that becomes complete instantaneously or by allowing instant
topological change to happen (see [Top11] for a nice survey). In higher dimensions the Kähler Ricci
flow has a similar behavior. In [SW11] Song and Weinkove describe carefully how the Kähler Ricci flow
contracts an exceptional divisor (creating a “forward bubble”) and converges to the original manifold.
This is part of a more general program outlined by Song and Tian [ST09] to show that the Ricci flow on
algebraic varieties performs algebraic surgeries and eventually carries out an analytical version of Mori’s
(conjectural) minimal model program (the geometrization program for Kähler manifolds mentioned
before). A higher dimensional analogue of reverse bubbling is the “extraction” of an exceptional set.
This happens for example with the expanders constructed in [FIK03] or in higher codimension when
the Kähler Ricci flow performs a flip as described in [SY12] (these are modelled on higher rank vector
bundle versions of the solitons in [FIK03] also described in [DW11]. At the singular time these are
projective cones.).
1.5. Kähler Ricci flow
It was shown in [Cao85] that the (normalized) Ricci flow deforms a Kähler metric on a compact Kähler
manifold with first Chern class c1 (M ) < 0 or c1 (M ) = 0 to a Kähler Einstein metric therefore reproving
the results of Aubin and Yau. In a sense Kähler Ricci flow is the generalization of Ricci flow on surfaces
in higher dimensions. Much like the Ricci flow on the sphere the case c1 (M ) > 0 has attracted the most
attention. If c1 (M ) is not trivial or definite the Kähler Ricci flow develops singularities. Its behavior
before and after these singularities has been studied intensively in recent years. While the Kähler Ricci
flow can be studied independently from general Ricci flow as questions that arise in Kähler geometry
have no analogy in Riemannian geometry, Kähler Ricci flow serves also as a good testing ground for
general Ricci flow. As mentioned before there is a promising theory under development that would allow
the Kähler Ricci flow to flow through singularities supporting Perelman’s conjecture mentioned before.
In chapter 5 of this thesis we will construct self-similar solutions of the Kähler Ricci flow coming out of
Ricci flat Kähler cones.
1.6. Asymptotically conical Ricci flat Kähler manifolds
The construction of nearby Ricci flat manifolds or minimal surfaces asymptotic to a given Ricci flat
or stationary cone complements the above mentioned results on the existence of self-similar solutions
coming out of such cones. In [Ilm95] it is shown that the Simons cones Cp,q with p+q ≤ 7 admit asymptotic smooth minimal surfaces. Kronheimer [Kro89a] showed the existence of so called gravitational
instantons, i.e. ALE hyperkähler manifolds. These manifolds are deformations of crepant resolutions
of quotients of C2 by finite subgroups of SU (2) which are completely classified by the McKay correspondence. Joyce [Joy00] proved that ALE Ricci flat Kähler metrics exist on any crepant resolution
(vanishing first Chern class) of a quotient Cn /Γ by a finite subgroup Γ ⊂ SU (n). More recently these
results were generalized for crepant resolutions of general Calabi Yau cones by van Coevering [vC10],
Goto [Got12], and their deformations by Conlon and Hein [CH12].
These results should be compared to our theorem 6 which roughly says that a resolution of a Ricci flat
Kähler cone with negative first Chern class admits an expanding Kähler Ricci solitons metric.
In particular there exist ALE hyper-Kähler structures on crepant resolutions of C2 /Γk , where Γk is a
6
1. INTRODUCTION
cyclic subgroup of SU (2) of order k. Such a cone is diffeomorphic to C2 /Zk where Zk is a cyclic subgroup of U (2) that acts diagonally on C2 and has a non-stationary solution of the Ricci flow as shown
in [FIK03].
1.7. Overview
This thesis is concerned with Ricci flows which are initially Riemannian cones and become smooth instantaneously. In particular we study Ricci flows starting from Ricci flat cones and Kähler Ricci flows
starting Ricci flat Kähler cones. The next chapters are organized as follows:
In chapter 2 we recall some notation and definitions in Kähler geometry and present some classical
results such as the famous Calabi Yau theorem. We also prove the following long time existence results for Kähler Ricci flow on non-compact Kähler manifolds (which follows almost directly from the
arguments in [Cha04] and [TZ06]1).
Theorem 1. Let (M, J, g) be a complete Kähler manifold with bounded geometry. Assume that KM is
non negative in the sense that there exists a hermitian metric
√ h on KM such that the associated curvature
¯ , for some bounded smooth F . Then
form Θh is bounded and non-negative with ρ(ω) + 2πΘh = −1∂ ∂F
the Kähler Ricci flow on M exists for all times.
We go on to study Ricci flat Kähler cones and closely related Sasaki Einstein manifolds. Apart from
the basic definitions some of these results are more recent or less classical so we include some proofs in
the appendix. These cones have been studied by various authors mostly to construct asymptotic Ricci
flat Kähler manifolds. And of course we use their results for our purposes. In particular, we build on
the fact that any Ricci flat Kähler cone (including the vertex) can be understood as a normal affine
algebraic variety with an isolated rational singularity (more precisely a Q-Gorenstein singularity - which
make Ricci flat Kähler cones very special among all Kähler cones which are generally not Q-Gorenstein)
which admits an equivariant resolution. By equivariant we mean that the holomorphic action of the
homotheties of the cone has a unique lift to the resolution.
The topic of chapter 3 are general Ricci flows starting from Riemannian cones (R+ × Σ, dr2 + r2 gΣ ) over
smooth compact manifolds Σ which become smooth instantaneously. In other words, they are Ricci
flows on a smooth manifold M such that (M, g(t)) converges to the cone in the Gromov Hausdorff sense
and smoothly away from some compact subset. For such flows we show that the quadratic curvature
decay is inherited for positive times from the initial cone metric.
Theorem 2. Let (M, g(t))t∈(0,T ] be a complete Ricci flow with bounded curvature starting from the
Riemannian cone (C, gC ). Then |Rm|g(t) decays quadratically. In other words, for each t ∈ (0, T ] we
have
sup |Rm|(x, t)dg(t) (p, x)2 < ∞.
x∈M
Moreover, for any k ∈ N
sup |∇k Rm|(x, t)dg(t) (p, x)2+k < ∞.
x∈M
The proof of this result uses Perelman’s pseudolocality theorem for non-compact manifolds in combination with Shi’s curvature estimates for the Ricci flow. This gives a rough estimate for the spatial
asymptotics of the Ricci flow which can improved if the initial cone is Ricci flat:
Theorem 3. Let (M, g(t))t∈(0,T ] be a complete Ricci flow with bounded curvature starting from the Ricci
flat cone (C, gC ). Then for each κ > 0 and t ∈ (0, T ]
sup t|Rc|(x, t)σ(x, t)−κ−1 eσ(x,t) < ∞,
x∈M
where σ(x, t) = dg(t) (p, x)2 /4t.
1This has also been observed by Lott and Zhang, see theorem 4.1 and theorem 5.1 in [LZ11]
1.7. OVERVIEW
7
In other words, (M, g(t)) approaches the asymptotic cone at infinity exponentially fast. In contrast,
the asymptotically conical Ricci flat manifolds mentioned in the introduction approach the asymptotic
Ricci flat cone only at polynomial rate.
In chapter 4 we study expanding gradient Ricci solitons. We recall some well known facts about Ricci
solitons and prove a special converse of the previous theorem: If the Ricci flow is self similar expanding
and has quadratic curvature decay then its initial condition is a cone. More precisely, the self expanding
Ricci flow (M, g(t)) converges to a Riemannian cone as t & 0, i.e. it is a Ricci flow starting from a cone
in the sense above2:
Theorem 4. Let (M, f, g) be an expanding gradient soliton such that for some point p ∈ M and for
each k ∈ N0
sup |∇k Rm|(x)dg (p, x)2+k < ∞
x∈M
then the associated Ricci flow (M, g(t)) converges to a Riemannian cone C as t & 0.
This means that expanding gradient Ricci solitons are (under the assumption of quadratic curvature
decay) always examples of smooth Ricci flows coming out of cones.
Chapter 5 contains the main result of this thesis. We return to the Kähler setting and prove that
if a Ricci flat cone admits a resolution which satisfies a certain negativity condition then there exist a
self-expanding Kähler Ricci flow coming out of the cone:
Theorem 5. Let (M, J) be a complex manifold biholomorphic via Φ to a Ricci flat Kähler cone
(C, JC , gC ) outside a compact subset. Suppose in addition that
(1) g is a complete Kähler metric on M such that Φ∗ g − gC = O(r−τ ), τ > 2, as r → ∞ and
(2) there exists a complete holomorphic vector field X on M such that
(a) L=(X) ωg = 0, where =(X) = −X + X
√
(b) Φ∗ (X) = 12 (r∂r − −1Jr∂r ) outside of a compact set
(c) <(X) = ∇θX , for some real valued θX .
Then, if there exists a smooth function F ∈ C ∞ (M ) such that
√
¯ + LX ωg ,
ρ(ωg ) + ωg = −1∂ ∂F
√
¯ such that
there exists a smooth and complete Kähler metric ωg0 = ωg + −1∂ ∂ϕ
ρ(ωg0 ) + ωg0 = LX ωg0
which is asymptotic to the cone at exponential rate.
We will see that in the case above M has to be a resolution of C and the assumption KM > 0 is
sufficient:
Theorem 6. Let (C, JC , gC ) be a Ricci flat Kähler cone. Suppose that π : M → C is an equivariant
resolution of C and X the holomorphic extension of the pull back the radial vector field across the
exceptional locus E. Then there exists an expanding (gradient) Kähler Ricci soliton metric on M with
vector field X if there exists a hermitian metric h on KM such that the associated curvature form Θh
is positive.
The theorem can be reduced to an elliptic Monge Ampere equation. We prove that a solution to this
equation exists and decays exponentially fast towards the cone metric.
Theorem 7. Let (M, J, g) be a Kähler manifold biholomorphic via Φ outside a compact set to C \ BR (o)
such that Φ∗ g = gC outside a possibly larger compact set. Moreover let X be a holomorphic vector field
on M such that
√
∂
∂
(1) Φ∗ X = 21 r ∂r
− −1Jr ∂r
outside a compact set.
2This is basically a corollary of similar result in [CD11] and we merely spell out some of the details of remark 1.5 in
[CD11].
8
1. INTRODUCTION
(2) <(X) = ∇θX for some real valued function θX on M .
(3) L=(X) ω = 0.
∞
Then for any F ∈ Cc∞ (M ) there exists a unique ϕ ∈ Cw
(M ) such that
(1) ωϕ > 0.
(2) ϕ satisfies the equation
ωϕn = eϕ−X(ϕ)+F ω n .
The proof of this theorem is very similar to the proof of the Calabi conjecture. The main difficulty
are the unbounded coefficients in the first order term. Here we will use global Schauder estimates of
Lunardi (theorem 9 below) which we will describe in the appendix.
We conclude this chapter by reviewing some known examples of asymptotically Ricci flat expanders and
present new examples which do not reduce to ordinary differential equations. We show:
Theorem 8. Let Γ be the subgroup of SU (2) generated by diag(e2πi/p , e2πiq/p ) for p, q ∈ N with p > q
and gcd(p, q) acting freely on C2 \ {0}. Then there exists an ALE expanding Ricci soliton metric on the
toric resolution of C2 /Γ if and only if for all integers (6= 1) in the continued fraction expansion of p/q
are strictly larger than 2. Moreover, these metrics approach the euclidean metric exponentially fast.
These resolutions can be seen as analogues of the resolutions considered by Kronheimer mentioned in
section 1.6. The corresponding Ricci flows converge to (C2 /Γ, geucl ) as t & 0.
In chapter 6 we discuss some open problems and further research directions. In particular, we mention
the expander version of Perelman’s entropy functional constructed in [FIN05] which might have applications for Ricci flows coming out of cones.
In the appendix we state and prove a version of a result of Lunardi [Lun98] for elliptic operators
with unbounded coefficients needed in chapter 5 which is suited for asymptotically conical manifolds.
Theorem 9. Let (M, g) be an asymptotically conical manifold and X the radial vector field outside a
compact set. Then for any f ∈ C 0,α (M ), α ∈ (0, 1) and µ > 0 there exists a unique u ∈ C 2,α (M ) such
that
∆u − X(u) − µu = f.
Moreover, there exists a uniform constant C = C(α, µ) such that
kukC 2,α (M ) ≤ Ckf kC 0,α (M ) .
We also remind the reader about some basic toric geometry in the second part of the appendix. In this
part we quote some maybe less known results for non-compact toric varieties.
CHAPTER 2
Preliminaries
2.1. Kähler Geometry
This chapter is meant to be a summary of material needed in the following chapters, in particular chapter
5. For general background on Kähler geometry and complex geometry see [Huy05], [Bal06], [GH94];
the Calabi Yau conjecture and related problems see [Aub98], [Bes08], [Tia00], [Joy00], and [Blo12];
Ricci flow see [CK04], [CCG+ 07],[CCG+ 08],[CCG+ 10], [CLN06]; Kähler Ricci flow [CCG+ 07],
[SW12], [BCG+ 12]; elliptic equations in general see [GT01], [Kry96], (Complex) Monge Ampere
equations in particular [PSS12] and [Aub98]; algebraic and complex analytic geometry see [GH94],
[KM98], [Law89], [Dem12]; for the minimal model program see [Mat02] and resolutions [Hau03];
for Sasaki geometry see [BG08] and [Spa11].
2.1.1. Complex manifolds. Let M be a smooth manifold of dimension 2n. The manifold M is
called a complex manifold if it can be covered by charts {Ui , ϕi } with homeomorphisms ϕi = (z1i , ..., zni ) :
Ui → Vi ⊂ Cn such that the transition maps ϕj ◦ ϕ−1
are biholomorphisms (a complex structure). This
i
∗
induces an almost complex structure
J
∈
Γ(T
M
⊗
T M ) with J 2 = −idT M on T M . For the local
√
holomorphic coordinates zj := xj + −1yj the endomorphism J is defined by
J
∂
∂
=
,
∂xj
∂yj
and J
∂
∂
=−
∂yj
∂xj
for j = 1, ..., n. In particular J 2 = −idT M . Conversely any even dimensional manifold M with an
almost complex structure J on M such that the Nijenhuis tensor
N (J)(X, Y ) := [X, Y ] + J([X, JY ] + [JX, Y ]) − [JX, JY ]
vanishes is a complex manifold in the sense that it admits a holomorphic coordinate charts such that
the induced almost complex structure is equal to J.
The almost complex structure J induces
bundle T M ⊗ C into
√ a splitting of the complexified tangent
√
an eigenspace T M 1,0 with eigenvalue −1 and T M 0,1 with eigenvalue − −1, respectively. Similarly
T M ∗ ⊗C = (T M 1,0 )∗ ⊕(T M 0,1 )∗ . In local holomorphic coordinates we can give bases over C for T M 1,0 ,
T M 0,1 , (T M 1,0 )∗ and (T M 0,1 )∗ respectively by
√
√
∂
1
∂
∂
∂
1
∂
∂
=
− −1
,
=
+ −1
∂zj
2 ∂xj
∂yj
∂ z̄̄
2 ∂xj
∂yj
and
dzj = dxj +
And setting
^p,q
√
T M ∗ :=
−1dyj ,
^p
dz̄̄ = dxj −
(T M 1,0 )∗ ⊗
^q
√
−1dyj .
(T M 0,1 )∗ .
we can split (even if N (J) 6= 0)
^l
TM∗ =
M ^p,q
T M.
p+q=l
The exterior derivative
d : Ωk (M ) := C ∞ (M, ∧k T ∗ M ) → Ωk+1 (M ) := C ∞ (M, ∧k+1 T ∗ M )
maps
Ωp,q (M ) → Ωp+1,q (M ) ⊕ Ωp,q+1 (M ),
9
10
2. PRELIMINARIES
where Ωk,l (M ) := C ∞ (M, ∧k,l T M ) if and only if J is integrable (N (J) = 0). In holomorphic coordinates
Ωp,q (M ) are the forms of the type
X
α :=
αIJ dzi1 ∧ ... ∧ dzip ∧ dz̄j1 ∧ ... ∧ dz̄jq .
|I|=p,|J|=q
And so in this case d splits as
¯
d = ∂ + ∂.
where
∂ : Ωp,q (M ) → Ωp+1,q (M )
and
∂¯ : Ωp,q (M ) → Ωp,q+1 (M ).
¯ = −∂ ∂.
¯ This observation allows for the definition
Since d2 = 0 it follows that ∂ 2 = 0, ∂¯2 = 0, and ∂∂
of the Dolbeaut cohomology group
ker ∂¯ : Ωp,q (M ) → Ωp,q+1 (M )
p,q
H (M, C) =
.
¯ p,q−1 (M )
∂Ω
And similarly the Bott-Chern cohomology
p,q
HBC
(M, C) :=
{α ∈ Ωp,q (M ) : dα = 0}
¯ p−1,q−1 (M )
∂ ∂Ω
2.1.2. Holomorphic line bundles and the canonical bundle. Holomorphic vector bundles
are complex vector bundles (i.e. vector bundles with complex vector spaces as fibers) over complex
manifolds whose total space carries a complex structure compatible with that of the base. The complex
vector bundle T M 1,0 of a complex manifold (M, J) is an example of a holomorphic vector bundle. More
precisely
Definition 2.1.1. A holomorphic vector bundle of rank r over a complex manifold (M, J) is a complex
manifold E together with a holomorphic map π : E → M with each fibre π −1 (x) having the structure
of an r-dimensional complex vector space such that M admits a covering {Ui , i ∈ I} of open sets and
biholomorphic maps ψi : π −1 (Ui ) → Ui × Cr which satisfy (π(x), ·) = π̃(ψi (x)) (where π̃ projects to
the first factor) and ψi restricted to each fibre π −1 (x) is C-linear. A holomorphic line bundle is a
holomorphic vector bundle of rank 1.
Remark 2.1.2. The tensor product of 2 holomorphic vector bundles is a holomorphic vector bundle.
Every holomorphic vector bundle E over a complex manifold (M, J) admits a hermitian metric h (a
hermitian inner product h(x) on each fibre that depends differentiably on x). Moreover, there exists a
unique connection D (called Chern connection) on a hermitian holomorphic vector bundle (E, h) such
that for any pair of local sections σ1 and σ2 of E we have d(h(σ1 , σ2 )) = h(Dσ1 , σ2 ) + h(σ1 , Dσ2 ) and
¯ i.e. for a local section σ and a vector field X on M we have DX σ = D1,0 σ + ∂σ(X)
¯
D0,1 = ∂,
=
X
0,1
¯
DX 1,0 σ + ∂σ(X
) where X = X 1,0 + X 0,1 are the parts of X in T M 1,0 and T M 0,1 respectively. This
connection gives rise to a curvature operator D2 on E and if r = 1 (a holomorphic line bundle) then the
curvature operator can be identified with a real closed 2-form which we will denote by
Θh on M (the
√
−1 ¯
curvature form of (E, h)). This curvature form can be locally expressed as Θh (x) = − 2π ∂ ∂ log(h(x)).
For a holomorphic line bundle the first Chern class c1 (L) is defined as the cohomology class in H 2 (M, R)
represented by Θh . One can check that this definition does not depend on the choice of h (See [GH94]
for details). For a complex manifold (M, J) the canonical line bundle is the holomorphic line bundle
^n,0
KM :=
TM∗
and the first Chern class of M is defined as
c1 (M ) := −c1 (KM ).
2.1. KÄHLER GEOMETRY
11
2.1.3. Kähler geometry. A hermitian structure on a complex manifold (M, J) is a Riemannian
metric g on M such that for any x ∈ M and X, Y ∈ Tx M
g(JX, JY ) = g(X, Y ).
The hermitian metric h on M is then the complex sesquilinear extension of g to T M ⊗ C which splits as
√
h(X, Y ) = g(X, Y ) − −1ω(X, Y ),
where ω(·, ·) = g(J·, ·) is a real (1, 1)-form. In local coordinates ω is of the form1
√
−1
n
X
hi̄ dzi ∧ dz̄ ̄ .
i,̄
In the following we will write gi̄ instead of hi̄ .
Definition 2.1.3. A Kähler structure on a complex manifold (M, J) is a hermitian structure h on
(M, J) such that dωg = 0. The metric g induced by the hermitian structure is then called a Kähler
metric and ωg a Kähler form. A complex manifold with a Kähler structure is a Kähler manifold.
The 2-form ωg is a representative of a cohomology class in H 1,1 (M, R) called Kähler class. Note that
the condition dωg = 0 togehter with the integrability of J implies that ∇J = 0 where ∇ is the associated
Levi Civita connection in T M . In holomorphic coordinates this means
∂gk̄
∂gi̄
=
,
∂zk
∂zi
∂g
∂gi̄
= ik̄ .
∂ z̄k̄
∂ z̄̄
We can extend the Levi Civita connection ∇ C-linearly to T M ⊗ C and so we obtain for the Christoffel
symbols in complex coordinates
∇∂zi ∂zj = Γkij ∂zk = g kl̄ ∂zi gj l̄ , ∇∂ z̄ı̄ ∂z̄̄ = Γk̄ı̄̄ ∂z̄k̄ = g lk̄ ∂z̄ı̄ gl̄ , ∇∂zi ∂z̄̄ = 0, ∇∂ z̄ı̄ ∂zj = 0 .
This shows that ∇ = ∇C can be split as ∇ = ∇1,0 + ∇0,1 where ∇1,0 is ∇C restricted to T M 1,0 and ∇0,1
0,1
is ∇C restricted to T M 0,1 , i.e. ∇1,0
Y X = ∇Y 1,0 X and ∇Y X = ∇Y 0,1 X. In the following we will write
¯ = ∇(0,1) for the (0, 1)-part of ∇ unless otherwise indicated.
∇ = ∇(1,0) for the (1, 0)-part of ∇ and ∇
Note that in particular for any smooth real valued function f
¯ ̄ f = ∇
¯ ̄ ∇i f
∇i ∇
does not depend on the choice of g. Moreover,
2
1
¯ = g i̄ ∇i ∇
¯ ̄ f = g i̄ ∂ f = n
∆g f = ∆∂¯f = ∂¯∗ ∂f
4
∂zi ∂ z̄̄
2
√
¯ ∧ ω n−1
−1∂ ∂f
g
.
ωgn
Remark 2.1.4. We will write ∆g (when g is a Kähler metric) for ∆∂¯ when acting on functions.
More generally, for a (1, 1)-form α
trg (α) = g i̄ αi̄ =
n α ∧ ωgn−1
.
2
ωgn
For the curvature tensor on T M ⊗ C associated to ∇ we have in local holomorphic coordinates
Ri̄kl̄ = Rm(ei , e̄ , ek , el̄ ) = −
∂gkp̄ ∂gml̄
∂ 2 gkl̄
+ g mp̄
∂zi ∂ z̄̄
∂zi ∂ z̄̄
where we wrote ej = ∂zj , and e̄ = ∂z̄j , j = 1, ..., n. And the complex Ricci curvature can be expressed
as
∂2
∂
kl̄
kl̄ ∂gkl̄
g
=−
log (det(gkl̄ )) .
Ri̄ = g Ri̄kl̄ = −
∂ z̄̄
∂zi
∂ z̄̄ ∂zi
P
1Note that with this convention ω
i
ı̄
n
geucl , the Kähler form associated to the euclidean metric h =
i dz dz̄ on C is of the
P
form ω = 2
n
i=1
dxi ∧ dy i .
12
2. PRELIMINARIES
It can be checked that (Ri̄ ) is a hermitian tensor on (M, J) and it splits into a real part Rcg (the Ricci
curvature of g) and an imaginary part ρg (·, ·) = Rcg (J·, ·), the associated Ricci form:
√
R(·, ·) = Rcg (·, ·) − −1ρg (·, ·).
In local coordinates
ρg =
√
−1
X
√ X
Rci̄ dzi ∧ dz = − −1
i,̄
√
i,̄
∂2
log(det(gkl̄ ))dzi ∧ dz̄̄ .
∂zi ∂ z̄̄
Abusing notation we will also write ρ(ω) = − −1∂ ∂¯ log(ω n ).
Lemma 2.1.5. Let h be a hermitian metric on the canonical line bundle KM of a Kähler manifold
∞
(M, J, g)
√ and Θh its curvature. Then there exists a real valued function f ∈ C (M ) such that 2πΘh +
¯
ρ(g) = −1∂ ∂f .
Proof. Using g we can define a hermitian metric h̃ on KM as follows. Let σ ∈ Γ(KM ) be a section
of the canonical line bundle. Then h̃ is defined by
σ ∧ σ̄
.
|σ|2h̃ :=
ωgn
One can check that the definition of h̃ does not depend on the choice of the local section σ and that
1
Θh̃ = − 2π
ρ(g). Now
√
−1 ¯
∂ ∂ log(h̃−1 h),
Θh − Θh̃ = −
2π
where f := log(h̃−1 h) is a well defined smooth real valued function on M .
¯
It follows that ρ(ω) is a representative of 2πc1 (M ). Moreover, even if the ∂ ∂-lemma
below does not
hold in general the difference between two Ricci forms is always exact in the Bott Chern cohomology.
2.1.4. Holomorphic vector fields.
Definition 2.1.6. Let (M, J) be a complex manifold. A real holomorphic vector field is a smooth
section ζ ∈ Γ(T M ) such that Lζ J = 0. A complex holomorphic vector field is a section X ∈ Γ(T (1,0) M )
¯ = 0.
¯ = ∂X
such that ∇X
√
Note that if X is a complex holomorphic vector field then it is of the form ζ − −1Jζ where ζ is a
real holomorphic vector field. Given a complex holomorphic vector field X we write ζ := <(X) and
ξ := =(X) = Jζ.
We will need to following two basic observations later:
Proposition 2.1.7. If M is simply connected, X a complex holomorphic vector field, and Lξ ωg = 0.
Then ζ is a gradient vector field: there exists a real valued function f such that X = (∇f )(1,0) .
Proof. Let e1 , ..., e2n be a local frame for T M . Since X is holomorphic we have Lξ J = 0 (and
Lζ J = 0) and so (using Einstein summation convention)
0 = hLζ Jei , ej i = h−∇Jei ζ + J∇ei ζ, ej i
= −Jik ∇ek ζj + Jjk ∇i ζk
and hence
−∇l ζj = Jli Jik ∇ek ζj = Jjk ∇i ζk Jli = ∇Jel Jζj .
On the other hand
0 = LJζ g(ei , ej )
= LJζ (g(ei , ej )) − hLJζ ei , ej i − hei , LJζ ej i
= h∇i Jζ, ej i + hei , ∇ej Jζi
= Jjk ∇i ζk + Jik ∇j ζk
= Jik ∇k ζj + Jik ∇j ζk
2.1. KÄHLER GEOMETRY
So ∇k ζj = ∇j ζk and since π1 (M ) = 0 it follows that ζ = ∇f .
13
Remark 2.1.8. Of course π1 (M ) = 0 is not necessary, in fact, H 0,1 (M, C) = 0 would suffice.
Conversely
Proposition 2.1.9. Suppose X = (∇f )(1,0) is a holomorphic vector field on M and f some real-valued
function on M . Then ξ = J∇f is a Killing vector field.
Proof. First note that
0 = (LJ∇f J)ej
= LJ∇f (Jej ) − JLJ∇f ej
= −∇Jej (J∇f ) + J∇j (J∇f )
l mn
k mn
= −Jjk Jm
g ∇k ∇n f + Jkl Jm
g ∇j ∇n f
l mn
= −Jjk Jm
g ∇k ∇n f − g ln ∇j ∇n f
which implies
−Jjn ∇i ∇n f = Jin ∇n ∇j f
And so we conclude
LJ∇f g(ei , ej ) = LJ∇f (g(ei , ej )) − g(LJ∇f ei , ej ) − g(ei LJ∇f ej )
= g(∇ei J∇f, ej ) + g(ei , ∇ej J∇f )
k mn
k mn
= gkj Jm
g ∇i ∇n f + gik Jm
g ∇ j ∇n f
= −Jjn ∇i ∇n f − Jin ∇j ∇n f
= 0.
¯
2.1.5. The ∂ ∂-Lemma.
Let f be a real valued function. Then we can define an exact real (1, 1)form by
2
√
√
¯ = −1 ∂ f dz i ∧ dz̄ ̄ .
−1∂ ∂f
∂zi z̄̄
The following shows that in the compact case the converse is also true: any exact real (1, 1)-form is of
the form described above.
¯
Lemma 2.1.10 (∂ ∂-Lemma).
Let (M, J, g) be a compact Kähler manifold. If α is an exact real (1, 1)form, i.e. α = dη, then there exists a real-valued smooth function f on M such that
√
¯
α = −1∂ ∂f.
Note that this is not true if M is non-compact as the following example shows:
Example 2.1.11. Let M = ∆(1, 2) ∪ ∆(2, 1) = {|z1 | < 1, |z2 | √
< 2} ∪ {|z1 | < 2, |z2 | < 1} ⊂ C2 . Let η
¯
¯
be a ∂-closed (0,√1)-form which is not ∂-exact. Then let α := −1∂η. Obviously α is exact (α = dη).
¯ then ∂(∂f
¯ − η) = 0 but also ∂(
¯ ∂f
¯ − η) = 0. Hence d(∂f
¯ − η). Since the domain
However if α = −1∂ ∂f
¯
is contractible there exists a function h such that dh = ∂f − η and in particular ∂h = 0. But this implies
¯ − h) which is a contradiction. One can check that
that η = ∂(f
η :=
(z̄2 − a)dz¯1 − (z̄1 − a)dz̄2
(|z1 − a|2 + |z2 − a|2 )2
√
for a ∈ (1, 2) does the job2.
2Note that any holomorphic function f on M extends to a holomorphic function on M 0 := ∆(2, 2) ∩ {|z z | < 2}.
1 2
(2.1)
14
2. PRELIMINARIES
Let (M, J) be a complex manifold (not necessarily compact). We call a class [α] ∈ H 1,1 (M, R) a Kähler
class if there exists a representative ω such that ω(·, ·) > 03. On a compact
√ complex manifold by the
¯ for some smooth real
¯
∂ ∂-lemma
two Kähler forms ω1 , ω2 in the same Kähler class [α] differ by −1∂ ∂f
valued function. And so the cone
√
¯
Hω := {ϕ ∈ C ∞ (M ) : ω + −1∂ ∂ϕ}
parametrizes the space of Kähler metrics in the same Kähler class [ω].
2.2. Notions of Positivity
In algebraic geometry there are different notions of positivity and all of them also play an important
role in Kähler Ricci flow and related elliptic problems. The first notion is “ample”: a holomorphic line
bundle over a complex manifold (M, J) is called ample if for some integer p, sufficiently large, H 0 (M, Lp )
gives an embedding of M into CP N . Kodaira showed that if M is compact4 this is equivalent to the
property that c1 (L) can be represented by a positive (1, 1)-form. In other words there exists a Kähler
metric ω on M such that [ω] = c1 (L). In particular if the (anti-)canonical line bundle is ample then
±c1 (M ) can be represented by a positive (1, 1)-form, i.e. c1 (M ) has a sign. If the anti-canonical line
bundle is ample then the manifold (M, J) is called a Fano manifold. Note that any representative of
c1 (L) ∈ H 1,1 (M.R) is also the curvature form of a hermitian metric on L. This is not true in general, for
¯
example if the ∂ ∂-lemma
fails in the non-compact case. There are several weaker notions of positivity:
a holomorphic line bundle is called nef if hc1 (L), Σi ≥ 0 for every compact complex curve Σ ⊂ M . Note
that an ample line bundle is automatically nef. If the canonical line bundle is nef then M is called
minimal. A good reference for different notions of positivity in algebraic geometry is [Laz04].
2.3. Monge Ampere Equations and the Calabi Yau theorem
One major achievement in Kähler geometry is the proof of the Calabi conjecture by Aubin and Yau
([Yau78], see also [Tia00] and [Joy00]): given a compact Kähler manifold (M, J, ω) and a representative β of c1 (M ) ∈ H 1,1 (M, R) one can always find a Kähler metric in the same Kähler class as ωg such
that its Ricci form is exactly 2πβ. In particular, if c1 (M ) = 0 there exists a (unique) Ricci flat Kähler
metric in every Kähler class of M . Yau’s proof consists of an existence result for an elliptic Monge
¯
Ampere equation which is derived as follows: given β ∈ √
c1 (M ) then by the ∂ ∂-lemma
there exists a
¯
¯
smooth real-valued function f such that 2πβ − ρ(ω) = −1∂√∂f . And again using the ∂ ∂-lemma
a
¯
solution in the same Kähler class will be of the form ωϕ := ω + −1∂ ∂ϕ, i.e. ρ(ωϕ ) = 2πβ. Subtracting
the previous equation we have as an equivalent equation
√
¯
ρ(ωϕ ) − ρ(ω) = −1∂ ∂f,
which, considering the local expression for ρ, becomes:
n
√
√
ωϕ
¯
¯
− −1∂ ∂ log
= −1∂ ∂f.
ωn
Now one observes that the left hand side is in fact a global potential and so one gets the Monge Ampere
equation
ωϕn = e−f ω n ,
(2.2)
where we absorbed the free constant in f (recall that f is determined up to a constant). Yau showed
that this equation always has a solution if M is compact. His proof works roughly like this. Fix f such
that
Z
Z
e−f ω n = V =
M
ωn .
M
Then consider the set S of t ∈ [0, 1] such that
ωϕn = e−tf ω n ,
3By ω > 0 we mean that the symmetric 2-tensor ω(x)(·, J·) is positive for every x ∈ M .
4If M is not compact positivity of L does not imply ampleness in general, see [Ohs79]
(2.3)
2.3. MONGE AMPERE EQUATIONS AND THE CALABI YAU THEOREM
15
has a solution. Obviously 0 ∈ S hence S 6= ∅. The idea is to show that S is open and closed in [0, 1].
This proves in particular that 1 ∈ S which means that there exists a solution to equation (2.3). The
linearization of the left hand side of the continuity equation 2.3) at a point t ∈ S is just ∆ϕ which is
an isomorphism between the function spaces in question. And so the implicit function theorem tells us
that S is open. The more difficult part of the proof is to show that S is closed. This requires uniform
estimates of solutions ϕt , t ∈ S, in C 2,α . Once one has these estimates a bootstrapping argument gives
all the higher order estimates and then it is an easy consequence of Arzela Ascoli that S is closed.
Yau’s original proof of these a priori estimates goes along these lines: a Moser iteration argument shows
uniform C 0 -estimates (using previously established L1 estimates). This first estimate is Yau’s main
contribution to the proof of the Calabi conjecture and so historically it is actually the last. It is a
surprising feature of the equation (or the proof) that one can skip the estimates for the gradient and go
ahead to prove estimates for ∆ϕt which imply by subharmonicity (with respect to ω) uniform estimates
¯ t . The proof relies on the uniform estimates in C 0 and uses the maximum principle. This part
for ∂ ∂ϕ
uses the compactness of M only in the sense that the bisectional curvature of the complex manifold
(M, J, ω) is bounded from below. Note that the non-mixed derivatives ∇2 ϕt are not estimated. The
estimate for ∆ϕt then gives also a uniform bound for ∇ϕt . In the original proof Yau then goes on to
¯ t are uniformly bounded. The proof is based on previous
show that the first covariant derivatives of ∂ ∂ϕ
work of Calabi and Nirenberg. This part of the proof can be adapted to the non-compact case with
almost no modifications. Together these estimates are sufficient to apply Schauder theory and obtain
the required C 2,α -estimates.
The Calabi-Yau theorem has several interesting consequences. For example, it implies that any connected Fano manifold M (a compact complex manifold with c1 (M ) > 0) is simply connected. Indeed,
choosing a Kähler metric such that its Ricci form is a positive representative of 2πc1 (M ), the Bochner
formula for holomorphic (p, 0)-forms, p ≥ 1, implies that hp,0 (M ) = 0 and hence h0,p (M ) = 0. By
Bonnet-Myers the fundamental group π1 (M ) is finite and by Riemann-Hirzebruch-Roch the universal
cover M̃ is compact and satisfies χ(M̃ , O) = |π1 (M )|χ(M, O). However, using the same arguments
P
for M̃ we have χ(M̃ , O) = p h0,p (M̃ ) = 1 which implies that |π1 (M )| = 1 and hence M is simply
connected (See [Bal06]).
Remark 2.3.1. Ever since Yau’s work appeared in 1978 various steps in the proof have been simplified.
Most notably Evans, Krylov, and Safonov developed a general theory of non-linear convex elliptic
equations. In particular, their methods allow to obtain interior C 2,α -estimates (for some α ∈ (0, 1))
for the solution ϕt once estimates for the second derivative are known. As mentioned above the C 2 estimates in Yau’s proof do not really give bounds on all second derivatives of ϕt , however one can
show that the theory of Krylov and Evans can still be applied to obtain uniform C 2,α -estimates. While
the C 3 -estimates of Yau are global the estimates using Krylov-Evans-Safonov theory are purely local
which seems to be helpful in more general settings. The proof of the C 0 -estimate was modified for more
general settings by Kolodziej [Kol98] using pluripotential theory.
Apart from the improvements of Yau’s proof which resulted from new developments in the theory of nonlinear elliptic PDEs a lot of new results for the complex Monge Ampere equation have been obtained.
First of all Yau and Tian ([TY90], [TY91]) studied the equation in the non-compact case, Cheng and
Yau ([CY80]) considered the Monge Ampere equation related to the existence problem of complete
Kähler Einstein metrics on domains in Kähler manifolds (i.e. with infinite boundary condition), and,
more recently, a whole theory of generalized solutions has been developed which deals with the Monge
Ampere equation when the right hand side is degenerate (i.e. vanishing at some points) or unbounded
(i.e. bounded in a weaker sense, for example in Lp ). The geometric motivation to study these problems is,
for example, the question whether a Kähler manifold admits a Kähler Einstein metric on the complement
of a divisor D with prescribed singular behavior near D (i.e. a Kähler Einstein metric with edges). Other
questions consider the Kähler Einstein problem in singular varieties or the existence of singular Kähler
Einstein metrics on compact complex manifolds whose canonical bundle satisfies a weaker positivity
condition (for example nef and big or just nef) than the one (ample) necessary for the existence of
smooth Kähler Einstein metric (see for example [Tsu88]).
16
2. PRELIMINARIES
2.4. Kähler Ricci flow
Hamilton ([Ham86]) showed that the Ricci flow
∂gij
= −2Rcij (g(t))
∂t
on a compact manifold preserves the holonomy group of the starting metric (in the sense that the
holonomy group of the evolving metric will be contained in the initial one in the future) and in particular
if the initial metric is Kähler (i.e. Hol(g) ⊂ U (n)) then the evolved metric g(t) will also be Kähler as long
as it exists5. For the noncompact case this was shown by Shi in [Shi89], [Shi97] under the assumption
of completeness and bounded curvature. So let ω(t) be the Kähler form of the Kähler Ricci flow g(t).
In terms of classes we have
[ω(t)] = [ω(0)] − 2πtc1 (M ).
If M is compact it was shown by Cao [Cao85] that the (renormalized) flow exists for all times if c1 (M )
is positive, zero, or negative and converges in the last two cases as t → ∞ to a Kähler Einstein metric
(i.e. ρ(ω) = λω, λ ∈ {0, 1}). As before the Ricci flow equation reduces to a scalar parabolic Monge
Ampere equation: let ωt be defined as
ωt := ω0 − tρ(ω0 )
for some initial Kähler metric ω0 with bounded curvature. Then the evolution
equation satisfied by the
√
¯ is
Kähler potential ϕ of the Ricci flow with respect ot ωt , i.e. ω(t) = ωt + −1∂ ∂ϕ
∂ϕ
= log
∂t
ωϕn
ω0n
+ c(t),
where c(t) only depnds in t. Conversely any solution of the equation above with c(t) = 0 and ϕ(·, 0) = 0
gives rise to a Ricci flow with initial condition ω0 . It was later shown by Tian and Zhang [TZ06] that
the flow exists on a compact manifold as long [ω(t)] is positive, so in particular, if KM is nef (i.e. c1 (M )
non-positive) the flow exists for all t ∈ [0, ∞). This can be generalized to the non-compact case in the
following way. Recall that here we cannot assume that a positive or negative representative of c1 (M ) is
given by the Ricci curvature of some Kähler metric.
Theorem 2.4.1. Let (M, J, g) be a complete Kähler manifold with bounded geometry. Assume that KM
is non-negative in the sense that there exists a hermitian metric h on KM such that the
√ associated
¯ , for
curvature form Θh is non-negative and bounded in all derivatives with ρ(ω) + 2πΘh = −1∂ ∂F
some bounded smooth F . Then the Kähler Ricci flow on M exists for all times.
Proof. Let
T := sup {τ : g(t) u g, |Rm|(t) < ∞, ∀t ∈ [0, τ ]} .
Assume that T < ∞. Now set ωt := ω + 2πtΘh . Obviously
[ωt ] = [ω] − 2πtc1 (M ). One can check that
√
¯ where ϕ(x, t) is a real valued function that
the Ricci flow starting with ω is of the form ωt + −1∂ ∂ϕ
solves the initial value problem
√
¯ n
(ωt + −1∂ ∂ϕ)
∂ϕ
= log
− F, ϕ(x, 0) = 0.
∂t
ωn
Taking one more derivative in t we get
∂2ϕ
∂ϕ
∂ϕ
= ∆0
+ 2πtrg(t) (Θh ) ≥ ∆0
,
2
∂t
∂t
∂t
5This was also discovered by Cao earlier in [Cao85].
2.5. SASAKIAN GEOMETRY
17
where ∆0 is the Laplacian with respect to the metric g(t) evolving by the Ricci flow. On the other hand
∂2ϕ
∂ϕ
= ∆0
+ 2πtrg(t) (Θh )
2
∂t
∂t
1
∂ϕ n 1
= ∆0
+ − trg(t) (ω) − ∆0 ϕ
∂t
t
t
t
n
∂ϕ 1
0
− ϕ + .
≤∆
∂t
t
t
And so
∂ϕ
∂ϕ
∂
t
− ϕ − nt ≤ ∆0 t
− ϕ − nt .
∂t
∂t
∂t
The parabolic maximum principle plus the known short time existence theorem gives a uniform bound
(only dependent on T and the initial conditions) for ϕ on M × [0, T̃ ] for all T̃ < T and hence we know
by the lemma below
kϕ( · , t)kC k,α (M ) < C(T, g, h, k, α),
for some constant C(T, g, h, k, α) which only depends on the initial condition, T , the curvature form
Θh , and k + α. But this means that the conditions of the theorem are satisfied even up to T and hence
beyond which is a contradiction. This proves that T = ∞.
Lemma 2.4.2. Let (M, J) be a complex manifold of complex dimension n and ωt = ω0 + tχ a smooth
path of complete and equivalent Kähler metrics with bounded geometry for all t > 0. Suppose that ϕt is
a smooth solution on M × [0, T ) of
√
¯ n
(ωt + −1∂ ∂ϕ)
∂ϕ
= log
+F
∂t
ω0n
for some function F ∈ ∩k C k,α (M × [0, T ]). Assume further that ϕ is uniformly bounded on M × [0, T ).
Then ϕ is uniformly (independent of t) bounded in C k,α (M ) for each k and α ∈ (0, 1).
Proof. The arguments are basically the same as in [Cha04] with the exception that ω is replace
by ωt . Indeed, let
A := log(n + ∆t ϕ) − κϕ,
where ∆t is the Laplacian with respect to ωt and κ ∈ R a constant depending on ωt . Then following
[Cao85] we can find uniform constants c1 = c1 (ωT , ω) and c2 (ωT , ω) such that
∂A
− ∆0 A ≤ −c1 eA/(n−1) + c2 .
∂t
It follows from the maximum principle that A is uniformly bounded on M by a constant C(ωt ). The
higher order estimates follow from the arguments in [Cha04] similarly. Note that in the non-compact
setting in order to apply parabolic Schauder estimates one needs in addition to the third order estimates
fourth and fifth order estimates as pointed out in [Cha04].
Remark 2.4.3. Theorem 2.4.1 also follows from theorem 4.1 in [LZ11]. The proof described above
is almost the same as in [LZ11] as it is based on the arguments in [TZ06] for the compact case and
[Cha04] which combined need almost no modification.
2.5. Sasakian Geometry
Definition 2.5.1. A Sasaki manifold is an odd-dimensional manifold S together with a quadruple
(η, ξ, Φ, g) (a Sasaki structure) consisting of
(1) a 1-form η such that η ∧ dη n−1 6= 0 (a contact form),
(2) a vector field ξ such that η(ξ) = 1 and ıξ dη = 0 (a Reeb vector field),
(3) an endomorphism Φ ∈ Γ(End(T M )) such that Φ2 = −1 + ξ ⊗ η,
(4) a Riemannian metric g such that g(Φ·, Φ·) = g(·, ·) − 41 η ⊗ η (compatibility) and (∇X Φ)Y =
g(X, Y )ξ − 14 η(Y )X (integrability).
18
2. PRELIMINARIES
It follows that Φ(ξ) = 0 and D := Ker(η) together with the restriction JD := Φ|D is a complex vector
bundle over S. Moreover, dη = 2g(,̇Φ·) and hence (D, JD , dη) is a stricty pseudoconvex CR structure
on S. The Reeb vector field induces a foliation Fξ of one dimensional leaves, i.e. each leaf is an
integral curve of ξ. The space of leaves which is the quotient space of this foliation is usually not a
manifold (see comments below). However, it can be given a so-called transverse complex structure as
follows. The manifold S can be covered by a foliation atlas {Uα , (xα , zα )}, xα : Uα → I ⊂ R where
zα : Uα → Vα ⊂ Cn−1 are surjective submersions such that
(1) ξ|Uα = ∂x∂α
(2) the transition maps zα ◦ zβ−1 : zβ (Uα ∩ Uβ ) → zα (Uβ ∩ Uα ) are biholomorphisms.
Since the Reeb vector field ξ is a Killing vector field (Lξ g = 0) the metric g induces a hermitian metric
gαT := (zα )∗ g|Uα on each patch Vα with the associated form ωαT . One can check now that zα∗ gαT is just
the restrction of g to D and zα∗ ωαT is equal to the restriction of 21 dη to {x ∈ Uα : xα = const}. And so
it follows in particular that ωαT is closed and hence gαT is Kähler. The collection {(ωαT , gαT )} = (ω T , g T )
is called a transversal Kähler structure on S. The transversal Kähler metric g T can be identified with
the restriction of g to D and hence the definition of the transverese Kähler structure does not depend
on the exact choice of holomorphic foliation atlas.
The picture becomes complete with the basic cohomology of the Reeb foliation. A p- form β on a Sasaki
manifold is called basic if Lξ β = 0 and ıξ β = 0. This condition implies that dβ is also basic. Indeed
ıξ dβ = Lξ β − dıξ β = 0
and
Lξ dβ = dıξ dβ = 0.
And so we have a cochain complex
d
d
d
d
d
B
B
B
B
B
Ω0B −→
Ω1B −→
.... −→
ΩpB −→
Ωp+1
−→
...
B
where dB is the restriction of d to the set of global sections ΩpB of the sheaf of basic p-forms. The
p
corresponding cohomology is called basic de Rham cohomology and denoted by HB
(S, ξ). Similar to
p
the complex case ΩB can be split as
M k,l
ΩpB =
ΩB
k+l=p
where a basic p-form β is of type (k, l) if for each chart (Uα , (xα , zα )) β is of the form
X
X
β=
βi1 ...ik j1 ...jl dzαi1 ∧ ... ∧ dzαik ∧ dz̄αj1 ∧ ... ∧ dz̄αjl .
i1 <..<ik j1 <...<jl
One can check that this definition does not depend on the exact choice of foliation atlas. This gives rise
to Dolbeaut operators
∂B
∂¯B
:
k+1,l
Ωk,l
B →Ω
:
k,l+1
Ωk,l
B →Ω
2
2
which satisfy ∂B
= 0 and ∂¯B
= 0. The associated basic Dolbeaut cohomology groups we denote by
p,q
H∂¯ (S, ξ). On each leave Vα we can define a transversal Ricci curvature RcTα and together with the
B
complex structure on Vα a transversal Ricci form ρTα . Just as in the Kähler case we have
√
ρTα = − −1∂ ∂¯ log(det(gαT )).
This gives a globally defined basic and closed (1, 1)-form ρT and a hermitian tensor RcT on S. The
1
B
basic cohomology class 2π
[ρT ] in H∂1,1
¯B is called basic first Chern class c1 (S, ξ) (each representative of
this class is also a representative of the first Chern class c1 (D) of the complex vector bundle D) also
denoted by c1 (Fξ ). Again similar to the Kähler case cB
1 does not depend on the choice transverse Kähler
metric, i.e. any Sasaki structure on S with a Reeb vector field proportional to ξ that gives the same
transverse complex structure (in particular the basic (Dolbeaut)-cohomology remains unchanged) will
also give back the same basic first Chern class. As for Kähler manifolds there exists a Sasaki analogue of
2.6. SASAKI EINSTEIN MANIFOLDS
19
1,1
¯
the ∂ ∂-lemma:
if γ and δ are two real closed basic (1, 1)-forms such that [γ] =√
[δ] in HB
(S) then there
exists a real valued basic function ϕ on S (i.e. ∇ξ ϕ = 0) such that γ = δ + −1∂B ∂¯B ϕ. Conversely,
fixing the transverse
complex structure and ξ, any deformation of the transverse Kähler form ω T of
√
the form ω T + −1∂B ∂¯B ϕ > 0, where ϕ is a basic function, gives rise to a Sasaki structure on M
with the same transverse complex structure and the same ξ. In fact, this Sasaki structure is unique
if one also requires that the complex structure on the cone is the same. This kind of deformation is
often called transverse Kähler deformation. The now obvious analog of the Calabi conjecture for Sasaki
manifolds was proven by El Kacimi-Alaoui [EKA90]: suppose β is basic real closed (1, 1)-form β is a
representative of 2πc1 (Fξ ). Then in any transverse Kähler class [ω T ] there exists a unique transverse
Kähler form, i.e. a transverse Kähler deformation ω̃ T of ω T , such that β is the transverse Ricci form ρ̃T
of ω̃ T .
Example 2.5.2. A first example of a Sasakian manifold is the round sphere S 2n−1 in Cn with the
induced Sasaki structure. This is not the only Sasaki structure on S 2n−1 . In fact, there are many other
Sasaki Einstein structure S 2n−1 (see [BG08]). More generally, for any finite subgroup Γ ⊂ U (n) that
acts freely on S 2n−1 the quotient S 2n−1 /Γ has a canonical Sasaki structure.
Definition 2.5.3 (Regularity). A Sasaki manifold is called regular if all orbits generated by ξ are
compact and of equal length. It is called quasiregular if all orbits are compact. And it is called irregular
if ξ has noncompact orbits.
It can be shown that a regular Sasaki manifold is a U (1)-bundle over a smooth Kähler manifold whereas
a quasi-regular Sasaki manifold is a U (1)-bundle over a Kähler orbifold. More precisely
Theorem 2.5.4 (from [BG08]). Let S be a compact regular (quasi-regular) Sasaki manifold. Then the
space of leaves of the Reeb foliation Fξ is a compact Kähler manifold (orbifold) (Z, ωZ , JZ ). And the
projection π : S → Z is a (orbifold) Riemannian projection, such that the fibres are totally geodesic
circles. The cohomology class [ωZ ] is proportional to an integer class in the (orbifold) cohomology
HZ2 (Z, Z).
Example 2.5.5. S 2n−1 with the standard Sasaki structure is a regular Sasaki manifold as a U (1) bundle
over CP n−1 . An example of a quasi-regular Sasaki manifold is S 3 /Γ where Γ ⊂ U (n) is a finite cyclic
subgroup that is not generated by a multiple of the identity. We will see later that a manifold can admit
several Sasaki structures. For example the circle bundle associated to the canonical line bundle over
CP 2 #2CP 2 admits an irregular Sasaki Einstein structure but not a regular one (see remarks below).
2.6. Sasaki Einstein manifolds
A Sasaki manifold (S, ξ, η, Φ, g) is called Einstein if its metric satisfies Rc(g) = (2n−2)g, where dim(S) =
2n − 1. This implies that RcT = 2ng T and one can check that the converse is also true. As in the
B
complex case the basic first Chern class is called positive (negative), cB
1 (Fξ ) > 0 (c1 (Fξ ) < 0), if
B
B
c1 (Fξ ) (−c1 (Fξ )) is represented by a transverse Kähler form. Given a Sasaki structure it is natural to
ask whether there exists a transverse Kähler deformation such that the new Sasaki structure is Sasaki
1,1
T
Einstein. A necessary condition is obviously that 2πcB
1 (Fξ ) = 2n[ω ] in HB (Fξ ), in other words,
B
c1 (Fξ ) > 0. Not surprisingly this problem is just as difficult as in the Kähler case. However, there is
a large number of examples (even irregular ones) which we will consider later. A straight forward way
to construct a Sasaki Einstein manifold is to start out with a Kähler Einstein manifold (Z, gZ , ωZ ) of
complex dimension n − 1 and Einstein constant 2n, e.g. CP n−1 with an appropriately scaled Fubini
Study metric, and a principal U (1)-bundle π : Σ → Z with first Chern class c1 (Σ) = c1 (Z)/I(Z), I(Z)
being the Fano index (the largest positive integer such that c1 (Z)/I(Z) is still an integral class). Then
the total space S of Σ together with the metric π ∗ gZ +η ⊗η is a regular simply connected Sasaki Einstein
manifold, where η is some connection of Σ with dη = 2π ∗ ωZ . Conversely any regular simply connected
Sasaki Einstein manifold S is of this form. Passing to universal covers (Sasaki Einstein manifolds always
have a finite fundamental group) regular Sasaki manifolds which are not simply connected can be treated
similarly. The quasiregular case is very similar and generalizes the above construction to orbifolds. This
shows for example that there does not exist regular Sasaki Einstein structure on principal U (1) bundles
20
2. PRELIMINARIES
over CP 2 #kCP 2 as above for k = 1, 2. However, there still exist irregular Sasaki Einstein structures in
both cases (see [FOW09]). For more details we refer to the survey of J. Sparks [Spa11].
2.7. Kähler Cones
A Riemannian cone (C(S), gC ) over a compact manifold Riemannian manifold (S, gS ) is called a Kähler
cone if there exists a compatible integrable almost complex structure J on C such that ξ := Jr∂r /2 ∈
Tr,s S ⊂ Tr,s C. The Euler vector field r∂r is a holomorphic vector field on C(S):
(Lr∂r J)X = [r∂r , JX] − J[r∂r , X]
C
= J∇gr∂Cr X − ∇gJX
r∂r − J∇gr∂Cr X + J∇gXC r∂r
C
= J∇gXC r∂r − ∇gJX
r∂r
C
r∂r
= J∇gXCt +Xr ∂r r∂r − ∇g(JX)
t +(JX)r ∂r
= JX − (JX)r ∂r − (JX)t
= 0,
where X is any vector field on C and (X)t and (X)r ∂r denote the part of X which is tangential to S
and the radial part of X respectively. The characteristic vector field ξ is also holomorphic and satisfies
in addition
Lξ gC = 0.
The Kähler structure on C(S) induces a Sasaki structure on S as follows: first let
√
η := 2 −1(∂¯ − ∂) log(r) = 2d log(r) ◦ J −1 .
Then one can check that η restricted to {1} × S satisfies η ∧ dη n−1 6= 0, and η(ξ) = 1. Moreover, Φ
can be defined on S by Φ := J|T S − η ⊗ ξ. It follows that Φ and gS satisfy the compatibility and the
integrability condition (see [BG08] for details). Altogether (η, ξ, Φ, g) gives a Sasaki structure on S.
Conversely a Sasaki structure (ξ, η, Φ, gS ) on S gives rise to a complex structure on the cone C(S) and
a Kähler structure defined by
2
2
√
¯ 2 = d(r η) = rdr ∧ η + r dη.
ωC := −1∂ ∂r
(2.4)
2
2
and
2
√
¯ 2 ( · , J · ) = rdr ∧ η( · , J · ) + r dη( · , J · )
gC := −1∂ ∂r
2
r2
= rdr( · )η(J · ) − rdr(J · )η( · ) + dη( · , J · )
2
= dr2 + r2 gS .
Theorem 2.7.1. [vC10] Let (C(S), JC , ωC , gC ) be the Kähler cone over a Sasaki manifold (S, Φ, η, ξ, g).
Then C(S) ∪ {o} is a normal complex analytic space, in fact it is an affine normal algebraic variety in
CN for some N > n.
Proof. See appendix B.
Remark 2.7.2. On a singular variety V the canonical line bundle KV still has a meaning. However, it
is not a holomorphic line bundle over V anymore. Details are described in appendix B. In the case of a
Kähler cone (or more generally a complex variety with sufficiently mild singularities) KC(S)∪{o} can be
defined as a more general object. More precisely, if L is a holomorphic line bundle over a non-singular
variety V then the sheaf of holomorphic sections OV (L) is what is called an invertible sheaf (a sheaf of
OV -modules that is locally isomorphic to OV , the sheaf of holomorphic functions on V , see appendix
B). Conversely, it is also true that every invertible sheaf F on a non-singular variety gives rise to a
holomorphic line bundle (i.e. F = OV (L)). On a singular variety invertible sheaves can still be defined
and are therefore the natural extension of the definition of holomorphic line bundles to (normal) singular
varieties. Invertible sheaves have the following additional property: Suppose L is an invertible sheaf on
a (normal) singular variety V . If π : Ṽ → V is a resolution then the inverse image of the sheaf π ∗ L
2.8. RICCI FLAT KÄHLER CONES
21
is also an invertible sheaf and hence the sheaf of holomorphic sections OṼ (π ∗ L) of a holomorphic line
bundle on Ṽ which we will denote by π ∗ L. If the singularities are for example Gorenstein then KV can
be understood as an invertible sheaf6. In case of a Ricci flat Kähler cone (see below) the singularities
are Q-Gorenstein ([vC10], see theorem B.2.1) which means that for some sufficiently large positive
p
integer p ∈ N the sheaf ı∗ OC(S) (KC(S)
), where ı : C(S) → C(S) ∪ {o}, is invertible. For such a cone
p
p
.
ı∗ OC(S) (KC
) replaces KC(S)∪{o}
2.8. Ricci flat Kähler cones
The Ricci curvature of an n-dimensional Riemannian cone (C(S), dr2 + r2 gS ) is given by
Rc(gC ) = Rc(gS ) − (n − 2)gS .
It follows that the cone (C(S), gC ) is Ricci flat if and only if Rc(gS ) = (n − 2)gS . In particular, Ricci
flat Kähler cones are Riemannian cones over Sasaki Einstein manifolds. The following proposition of
van Coevering gives a necessary condition for C(S) to admit a Ricci flat Kähler cone metric.
Proposition 2.8.1 ([vC10]). If C(S) is a Ricci flat Kähler cone then for some positive integer√p the p-th
p
power of the canonical line bundle KC(S)
admits a nowhere vanishing section Ω with Lξ Ω = −1npΩ.
We call C(S) a Calabi Yau cone if KC(S) (i.e. p = 1) admits a global non-vanishing section. This implies
in particular KC(S)∪{o} := ı∗ OC(S) (KC(S) ), where ı : C(S) → C(S) ∪ {o}, is not only an invertible sheaf
but also trivial or, equivalently, it admits a global non-vanishing section (it follows that C(S) ∪ {o}
is Gorenstein, which is the case if S is simply connected). The holonomy group of (C(S), JC , gC ) is
contained in SU (n) if and only if C(S) is Calabi Yau.
Example 2.8.2. Let Γ be a finite subgroup of U (n) acting freely on Cn \ {0}. Then S 2n−1 /Γ can be
given a canonical Sasaki Einstein structure (the standard one). And the corresponding Kähler cone
(C(S), gC ) := (Cn /Γ, geucl ) is Ricci flat (in fact, it is flat). The canonical divisor is not Cartier (see
Appendix B) in C(S) ∪ {o} (i.e. the sheaf ı∗ OC(S) (KC(S) ) is not invertible) in general. However, if
Γ ⊂ SU (n) (or more generally Γ ⊂ Sl(n, C)) then the quotient Cn /Γ is indeed Calabi Yau in the sense
above.
2.8.1. Resolution of Ricci flat Kähler cones. Since C(S) ∪ {o} is a normal algebraic variety
it can, by Hironaka’s theorem (see [EV03]), always be resolved and so for any given Kähler cone
there exists a smooth complex manifold M which is biholomorphic to C outside a compact set7. Since
H 1 (S, R) = 0 (S has positive Ricci curvature) we know that ı : Hc2 (M, R) → H 2 (M, R) is a well defined
injection and if we assume that the Kähler metric ω on M in the Kähler class [ω] ∈ H 2 (M, R) (if it
exists) approaches the cone metric sufficiently fast then [ω] ∈ ı(Hc2 (M, R)).
Proposition 2.8.3. Let π : M → C(S) ∪ {o} be a resolution of a Kähler cone C(S). Suppose ω is a
Kähler form on M such that ωC − π∗ ω = O(r−τ ), τ > 2. Then [ω] ∈ ı(Hc2 (M )).
Proof. Let Σ be a compact 2-cycle in M \ K. Then Σ is a 2-cycle in S × {1} and by Stokes’
theorem
Z
Z
π∗ ω − ωC =
π∗ ω − ωC = AreagS (Σ)(r2−τ ) → 0, as r → ∞
Σ×{1}
Σ×{r}
hence [π∗ ω − ωC ] = 0 in H 2 (C, R) and so [ω] ∈ ı(Hc2 (M, R)).
It follows that any Kähler metric on M asymptotic to the cone can be found in ı(Hc2 (M, R)). Similarly
one can prove that c1 (M ) ∈ ı(Hc2 (M, R)). From now on we will write Hc2 (M, R) for ı(Hc2 (M, R)).
6If it is clear what is meant we will write K for the invertible sheaf.
V
7The converse is also true: If we assume that M is biholomorphic to C(S) outside a compact set then this biholomorphism
φ : M \ K → C(S) \ B(o, R) can be extended across the compact set K to become a resolution φ̃ : M → C(S) ∪ {o} of the
complex space C(S) ∪ {o} described above (see [vC12]).
22
2. PRELIMINARIES
Remark 2.8.4. As we have seen above the cone C(S) ∪ {o} can be understood as an affine variety
V := C(S) in CN with an isolated singularity at the origin (even if C(S) is not Ricci flat). The radial
holomorphic vector field is then a tangent vector field on the regular part Vreg of V . Moreover, it extends
to a holomorphic vector field on CN . Namely, it corresponds to the action of C∗ : CN → CN , zj 7→ eaj t zj ,
aj ∈ R, t ∈ C. Now this group action of C∗ has a unique lift to a minimal resolution M of the singular
variety by theorem 1.5 in [EV03]. We will call M an equivariant resolution. This gives rise to a
holomorphic vector field on the nonsingular variety M which is a lift of the original radial holomorphic
vector field. In other words a Kähler cone always has a minimal resolution which admits a unique lift
of the radial holomorphic vector field, i.e. an equivariant resolution.
Proposition 2.8.5. Let π : M → C(S) ∪ {o} be an equivariant resolution of a Ricci flat Kähler cone,
X a holomorphic vector field on M and ω a Kähler form on M . Then X is a gradient vector field with
respect to ω meaning that there exists a complex valued function θX such that grad1,0
g θX = X. Moreover,
if ξ := =(X) is a Killing vector field (Lξ g = 0) then θX is real valued.
Proof. Note that α := ıX ω is a holomorphic (0, 1)-form.
And by the arguments in the proof of
√
¯ X for some complex-valued function. If
proposition B.2.2 H∂1¯ (M ) = 0 which implies that α = −1∂θ
√
¯ X ) = 0 and hence (up to a pluriharmonic function) θX can be chosen
Lξ ω = 0 then =(dα) = =( −1∂ ∂θ
to be real-valued.
Proposition 2.8.6. [vC10] Suppose that π : M → C ∪ {o} is a resolution of a Ricci flat Kähler cone
C. Let E√:= π −1 (o) be the exceptional set and let ω1 be a complete Kähler metric on M such that
¯ ∈ A1,1 (M, C) for some smooth function f ∈ C ∞ (M, R) and β ∈ A1,1 (M, C) a closed
ω1 = β + −1∂ ∂f
(1, 1)-form with compact support. Then there exists a complete Kähler metric ω0 on M and compact
subsets K1 ⊂ K2 ⊂ M such that ω0 = ω1 on K1 and ω0 = π ∗ ωC on M \ K2 .
Proof. See appendix B.
Corollary 2.8.7. Let π : M → C ∪ {o} be an equivariant resolution of C ∪ {o}. And suppose ω1 and
β are as above with the additional assumption that Lξ β = Lξ ω1 = 0. Then there exists a Kähler metric
ω0 such that ω0 = ω1 inside a compact set and ω0 = π ∗ ωC outside a large compact set and Lξ ω0 = 0.
Let G be the closure of the one-parameter subgroup generated by ξ in Aut0 (M ) (The connected component of the identity). This is a compact subgroup and we claim that in any Kähler class [α] we can
find a Kähler form ω which is G-invariant. Indeed, let ω0 be an arbitrary Kähler form in [α] then φ∗g ω0
lies in the same Chern-Bott class as ω by the lemma below and since G is compact we can average over
G to obtain the desired invariant Kähler form.
√
¯
Lemma 2.8.8. Let φg be a an element of the identity component of Aut(M ). Then φ∗g ω = ω + −1∂ ∂f
for some real valued function f .
Proof. Let φt , t ∈ [0, 1] be a smooth path in G connecting
√ the identity with g and let ζ := ζg be
the corresponding holomorphic vector field. Then Y :=
ζ
−
−1Jζ is a complex holomorphic vector
√
¯ Y for some complex valued function θY .
field on M . And since H∂1¯ (M, C) = 0 we have ıY ω = −1∂θ
√
¯ Y − θY ) = −2 −1∂ ∂=(θ
¯
Similarly ıȲ ω = ∂θY and so Lζ ω = ∂ ∂(θ
Y ). And so
Z 1
Z 1
√
φ∗g ω − ω =
φ∗t Lζ ω dt = −2 −1∂ ∂¯
=(θY ) ◦ φt dt.
(2.5)
0
0
It follows that in each Kähler class Hc2 (M, R) we can find a Kähler metric ω which is G-invariant and
proportional to the pull-back of the cone metric outside of a compact subset.
2.8.2. 2-dimensional Ricci flat Kähler cones. If dimC (C) = 2 and Rc(gC ) = 0 then Rc(gS ) =
2gS and so S is a 3-dimensional space form: a quotient of a round sphere by a finite subgroup Γ of
SO(4) and C is of the form R4 /Γ. Since the standard Sasaki structure is the only Sasaki Einstein
structure on S 3 /Γ it follows that C = C2 /Γ and Γ ⊂ U (2) if we assume that C is a Ricci flat Kähler
2.9. ASYMPTOTICALLY CONICAL MANIFOLDS
23
cone. There are basically two different ways to obtain a smooth Kähler manifold asymptotic to C (see
next section for the definition of asymptotically conical). The first method is to resolve the singularity
by adding in an exceptional divisor resulting from a blow up. Such a resolution always exists and
is unique. The second method is to deform the singularity to a smooth manifold (this procedure is
called smoothing). See [Joy00] for a definition of deformation. Both these methods have been studied
extensively for Γ ⊂ SU (2). Such singularities are called rational double points and are completely
classified. They are commonly known as ADE-singularities (A for Ak , D for Dk , E for E6 , E7 , and E8
which are the Dynkin diagrams of the corresponding subgroups of SU (2)) and the intersection matrix of
the rational curves that make up the exceptional set is completely determined by this combinatorial data.
A minimal resolution of such a singularity always admits an ALE hyperkähler structure as shown in
[EH79], [GH79], [LeB91], [Hit79], [Kro89b], [Kro89a], and [Joy00]. In particular, these resolutions
are ‘crepant’ meaning that the canonical line bundle is trivial. These type of singularities also admit
smoothings which are (always) diffeomorphic to the corresponding minimal resolutions. In [Kro89a] it
is shown that these manifolds admit ALE hyperkähler structures as well. If Γ is a subgroup of U (2)
which is not contained in SU (2) then C2 /Γ does not admit a crepant resolution meaning that the unique
minimal resolution has a non-trivial canonical line bundle. If Γ is a cyclic subgroup of U (2) then the
minimal resolution is a toric resolution. Again the intersection matrix of such a minimal resolution can
be determined by the combinatorial data of Γ and it is shown in [CS04] that each such resolution admits
a scalar flat ALE Kähler metric. Ricci flat Kähler metrics on smoothings of C2 /Γ, Γ ⊂ U (2), which
are not necessarily hyperkähler, have been studied in [Şuv12]. The only such singularities that admit
a smoothing are cyclic (not necessarily of type Ak ) and the rational double points (ADE) mentioned
above. The smoothings of these cyclic singularities are not simply connected and hence in particular not
diffeomorphic to the corresponding toric resolutions (which are simply connected). Note that in complex
dimension greater than 2 there do not exist any non-trivial deformations of Cn /Γ, for Γ ⊂ U (n), (Cn /Γ
is rigid) [Sch71] and so for n ≥ 3 the ALE Kähler manifolds can assumed to be resolutions (see [Joy00]).
2.8.3. Higher dimensional Kähler cones. The class of 5 and higher dimensional Sasaki Einstein manifolds is much larger. Higher dimensional Ricci flat Kähler cones are not necessarily quotient
singularities. In [Joy00] Joyce investigates the case n ≥ 3 when C = Cn /Γ, Γ ⊂ SU (n). He shows that
on any crepant resolution M of Cn /Γ there exists a unique ALE Ricci flat Kähler metric in each Kähler
class8. Since Cn /Γ is rigid this is indeed the only interesting case to study. These results have been
generalized crepant resolutions of Calabi Yau cones by van Coevering [vC08], [vC09], [vC10], [vC11],
Goto [Got12], Santoro [San09], and deformations of Calabi Yau cones in [CH12].
2.9. Asymptotically conical manifolds
In this section we define asymptotically conical Riemannian and in particular asymptotically conical
Kähler manifolds. We start out with an arbitrary smooth Riemannian cone C := R+ × S over a
compact Riemannian manifold (S, gS ) with the cone metric gC := dr2 + r2 gS .
Definition 2.9.1. A Riemannian manifold (M, g) is called asymptotically conical to (C, gC ) if there
exists a compact subset K ⊂ M , a positive number R and a diffeomorphism Φ : M \ K → C \ BR (o)
such that for k ∈ N
sup
|∇k (Φ∗ g − gC )(x)|gC = O(fk,0 (r)), as r → ∞,
(2.6)
x∈C\Br (o)
where ∇ is the Levi Civita connection with respect to gC , and for α ∈ (0, 1)
k
∇ (Φ∗ g − gC )(x) − ∇k (Φ∗ g − gC )(y)
= O(fk,α (r)) as r → ∞,
sup
sup
d(x, y)α
x∈C\Br (o) y,d(x,y)<δ
(2.7)
where δ is the injectivity radius of gC at x, ∇k (Φ∗ g − gC )(x) − ∇k (Φ∗ g − gC )(y) is defined via parallel
transport along the geodesic joining x and y, and limr→∞ fk,α (r) = 0 for k ∈ N0 , α ∈ [0, 1), and r > R.
8This also follows from the results of Tian and Yau in [TY90] as Joyce points out.
24
2. PRELIMINARIES
Remark 2.9.2. Typically fk,α (r) = r−τ −k−α for some τ > 0. And if in addition (S, gS ) = (S n /Γ, gS n )
for some finite subgroup Γ ⊂ SO(n) acting freely on S n then (M, g) is called ALE (asymptotically
locally euclidean). Such manifolds have been studied extensively in various contexts. Here we are mostly
2
2
interested in the case where τ > 2 and the case where fk,α (r) = rk+2+α e−r /2 (or fk,α (r) = Ck e−µr /2 ,
µ > 0). In the first case we say that (M, g) is asymptotic to (C, gC ) at rate τ while in the second case
we say that (M, g) is asymptotic to (C, gC ) at exponential rate.
We also define the following function spaces:
k,α
Definition 2.9.3. Let (M, g) be an asymptotically conical manifold. And let Cloc
(M ) the space
of k times continuously differentiable functions with α-Hölder continuous k-th derivative on M . Let
w : M → R+ be some smooth weight function then
k,α
k,α
Cw
(M ) := {u ∈ Cloc
(M ) : kukCwk,α (M ) < ∞},
(2.8)
where
kukCwk,α (M ) :=
k
X
sup |∇j (wu)(x)| + sup
j=0 x∈M
sup
x y,d(x,y)<δ
|∇k (wu)(x) − ∇k (wu)(y)|
,
d(x, y)α
where δ is the injectivity radius of gC at x and ∇k (wu)(x) − ∇k (wu)(y) is defined via parallel transport
along a geodesic joining x and y.
Remark 2.9.4. Note that these function spaces are different from the ones that are usually defined in
order to study the Calabi Yau problem on AC manifolds. In chapter 5 w will be exp(θX ).
2.9.1. Asymptotically conical Kähler manifolds.
Definition 2.9.5. A Kähler manifold (M, J, g) is complex asymptotically conical to a Kähler cone
(C, JC , gC ) if (M, g) is asymptotic conical to (C, gC ) and in addition
sup
|∇k (Φ∗ J − JC )(x)|gC = O(gk,0 (r)), as r → ∞,
x∈C\Br (o)
and for α ∈ (0, 1)
sup
sup
x∈C\Br (o) y,d(x,y)<δ
k
∇ (Φ∗ J − JC )(x) − ∇k (Φ∗ J − JC )(y)
= O(gk,α (r)), as r → ∞,
d(x, y)α
(2.9)
where δ is the injectivity radius of gC at x, ∇k (Φ∗ g − gC )(x) − ∇k (Φ∗ g − gC )(y) is defined via parallel
transport along the geodesic joining x and y, and limr→∞ gk,α (r) = 0 for k ∈ N0 , α ∈ [0, 1), and r > R.
Definition 2.9.6. A Kähler manifold (M, J, g) is called asymptotic to a Kähler cone (C, JC , gC ) if
gk,α = 0, in other words Φ is a biholmorphism.
One can show that in the second case M is a resolution of C and [ωg ] ∈ Hc2 (M, R) under sufficient decay
assumptions on gk,α . Moreover, the first definition reduces to the second one in case M is ALE.
CHAPTER 3
Ricci Flows attaining the Cone as an initial condition
3.1. Introduction
In general the Ricci flow develops singularities meaning that there are regions where the curvature blows
up. Typically such a singularity is of conic type (see [FIK03]). In order to continue the flow one has to
understand the Ricci flow with more general metric spaces as initial conditions. Such flows have been
studied by Miles Simon. In [Sim09] Miles Simon studies Ricci flows of metric spaces which are Gromov
Hausorff limits of sequences of smooth and complete manifolds which are non-collapsed and whose Ricci
curvature is bounded from below. In the case dim(M ) = 3 he shows that for any such limit (X, d) there
exists a smooth Ricci flow (M, g(t)), t ∈ (0, T ] such that (M, g(t)) converges to the initial metric space
in the Gromov Hausdorff sense as t & 0. In particular it follows that M is homeomorphic to X and
hence X is actually smooth with a non-smooth metric at t = 0. In higher dimension this is no longer
the case as the following example shows
Example 3.1.1 (Eguchi Hansen). Let M := L−2 → CP 1 be the total space of the canonical line bundle
of CP 1 equipped with the Ricci flat Eguchi Hansen metric g. Then (M, x, gj = j −1 g), j ∈ N, x ∈ CP 1
is a non-collapsed sequence Ricci flat manifolds and the associated Ricci flows gj (t) = gj converge to a
cone (C2 /Z2 , δ) as j → ∞. In other words, the limit is not smooth.
Simon constructs Ricci flows with metric initial conditions by taking the limits of the Ricci flows on each
manifold (Mi , gi (t)) in the sequence. While the limit might not be smooth in higher dimensions even
for positive times there still might be a smooth Ricci flow with (X, d) = limi→∞ (Mi , gi ) as an initial
condition:
Example 3.1.2 (4-dimensional expanders in [FIK03]). Let Mk , k > 2, be a crepant resolution of
a complex 2-dimensional cyclic quotient singularity equipped with a ALE hyper-Kähler metric g (see
[Kro89a]). Again the sequence (Mk , x, gj = j −1 g) is non-collapsed and Rc(gj ) = 0, j ∈ N. The limit
is a cone (C2 /Zk , geucl ). This cone is isometric (although not biholomorphic) to the initial cones in
[FIK03]. But there it is shown that there exist smooth flows (M̃k , g(t)) which converge to the original
cones as t → 0. However, the Ricci curvature is not bounded from below for these flows as t & 0.
Now assume that (M, g(t)) is a smooth Ricci flow which has a metric space (X, d) with conical singularities as an initial condition. This means that (M, g(t)) converges in the Gromov Hausdorff sense to
(X, d) as t → 0 and smoothly away form the singular set to a smooth incomplete Riemannian manifold
(X \ Xsing , gX ) such that (X, d) is the metric completion of (X \ Xsing , dgX ). Let λj be a sequence of
positive real numbers converging to 0. Then, ideally, the blow up limit (M, xj , λ−1
j g(λj t)) will converge
to a smooth Ricci flow coming out of a cone (see next section for the definition).
Example 3.1.3 (compact Ricci flows with singular initial condition). Let Mk := P (L−k
CP 1 ⊕ C) be the
Hirzebruch surfaces with k > 2. Then one can construct a sequence of cohomogeneity one (rotationally
symmetric) Ricci flows on Mk such that the initial conditions converge to a compact singular metric
space which is obtained by the contraction of the zero section of the CP 1 -bundle that is Mk . However,
for positive times the injectivity radius and the curvature can be controlled and so one obtains in the
limit a smooth Ricci flow with a singular initial condition.
Another way to look at the problem of Ricci flows of cones is to consider only the smooth part of the
cone (C, gC ) as an incomplete Riemannian manifold with boundary and to require the Ricci flow to
25
26
3. RICCI FLOWS ATTAINING THE CONE AS AN INITIAL CONDITION
become complete instantly while allowing for topological change. The following related examples have
been constructed by Giesen and Topping [GT10]
Example 3.1.4 (instant completeness,[GT10]). Let D be the open unit disc in R2 with the euclidean
metric g0 . Clearly (D, g0 ) is not complete. Now let gj be a sequence of complete metrics on g with
negative curvature that converges to the euclidean metric on compact subset of D. Taking the associated
Ricci flows gj (t) on D it is shown in [GT10] that these flows (D2 , gj (t)) converge smoothly for t > 0
and the limit is a instantaneously complete Ricci flow g(t) which converges to g0 as t & 0 on compact
subset of D2 .
Example 3.1.5 (instant topological change, [GT10]). Let T 2 be the flat torus and p ∈ T 2 . Then
M := T 2 \ {p} can be equipped with a complete metric g with constant curvature −1. In [GT10] the
authors construct a sequence of metric gj on T 2 which converges on compact subset of T 2 \ {p} to g.
They show that the sequence of associated Ricci flows gj (t) converge smoothly for positive times giving
a limiting Ricci flow on T 2 which converges to g as t & 0 on compact subsets of T 2 \ {p}. This Ricci
flow changes the topology instantly by adding in the point {p}.
The following example shows how this applies to Ricci flows coming out of cones.
Example 3.1.6 (FIK solitons). The examples constructed in [FIK03] combine both of these phenomena.
Here the initial manifold can be seen as an incomplete smooth manifold with boundary. The smooth
forward flows constructed in [FIK03] are instantly complete and topologically different from the initial
cone as they add in a CP n−1 .
Other Ricci flows coming out of cones have been constructed in [Cao85], [GK04], [DW11], [SS10].
Here we first try to give an appropriate definition of such a Ricci flow. This definition may seem a
bit ad hoc which is due to the lack of a general definition for a weak Ricci flow. Then we go on to show
that the quadratic curvature decay of the cone as r → ∞ is preserved under the Ricci flow. This implies
in particular that the Ricci curvature of the evolving metric decays exponentially to 0 if the initial cone
is Ricci flat.
3.2. Ricci flows coming out of cones
Let (C(S), gC ) be a Riemannian cone over a compact Riemannnian manifold S which is smooth except
at the vertex o. Assume that there exists a smooth manifold M and a family of smooth and complete
Riemannian metrics {g(t) : t ∈ (0, T ]} such that
(1) there exists a diffeomorphism ϕ : C → M \ K where K is a compact subset of M ,
(2) the metric spaces (M, p, dg(t) ) converge to (C ∪ {o}, o, dgC ) in the pointed Gromov Hausdorff
topology, for some p ∈ K,
(3) ϕ∗ g(t) converges smoothly to gC away from the vertex,
(4) for each t ∈ (0, T ] the Riemannian curvature of g(t) is bounded on M ,
(5) g(t) satisfies the Ricci flow equation
∂g(t)
= −2Rc(g(t)),
∂t
then we say that (M, g(t)) is a complete Ricci flow with bounded curvature starting from the Riemannian
cone (C, gC ). If in addition (M, J, g(t))t∈(0,T ] is a Kähler Ricci flow, (C, JC , gC ) is a Kähler cone, and
ϕ is a biholomorphism then we say that (M, J, g(t)) is a Kähler Ricci flow coming out of the cone. To
prove the main theorem of this section we will need Perelman’s famous pseudolocality theorem:
Theorem 3.2.1 (Perelman’s Pseudolocality Theorem, [Per02]). Let (M, g(t)), t ∈ [0, T ] be a complete
Ricci flow with bounded curvature. There exist (n) (depending only on the dimension n of M ) such
that the following is true. Suppose that for some x0 ∈ M
Volg (B(x0 , r0 )) ≥ (1 − )ωn r0n
3.2. RICCI FLOWS COMING OUT OF CONES
27
where ωn is the volume of the Euclidean n-ball. Then, if
|Rm|(x)| ≤
1
,
r02
∀ x ∈ Bg(0) (x0 , r0 )
it follows that
|Rm|(x) ≤
1
,
(r0 )2
∀ (x, t) ∈ Bg(t) (x0 , r0 ) × [0, min{T, (r0 )2 }].
Proof. Due to Perelman [Per02] and in the non-compact version in [CTY11].
In combination with Shi’s curvature estimates we have the following
Corollary 3.2.2. In addition to the assumptions above suppose that
|∇k Rm|(x) ≤
1
r02+k
for all x ∈ B(x0 , r0 ) then there exists a Ck > 0 such that
|∇k Rm|(x, t) ≤
Ck
,
r02+k
∀ (x, t) ∈ Bg(t) (x0 , r0 ) × [0, min{T, (r0 )2 }]
Proof. See [Top10].
Theorem 3.2.3. Let (M, g(t))t∈(0,T ] be a complete Ricci flow with bounded curvature starting form
the Riemannian cone (C, gC ). Then |Rm|g(t) decays quadratically. In other words, let r(x) be the
distance function on C to the vertex. Then there exist constants C and λ independent of t such that for
(x, t) ∈ M \ K × (0, T ] with r(x)2 /4t > λ
|Rm|(x, t) ≤ Cr(x)−2 .
Moreover, for each k ∈ N there exists a constant Ck such that
|∇k Rm|(x, t) ≤ Ck r(x)−2−k .
for r(x)2 /4t > λ.
Remark 3.2.4. The decay property can be formulated as
sup |Rm|(x, t)dg(t) (p, x)2 < ∞
x∈M
for all t ∈ (0, T ] and
sup |∇k Rm|(x, t)dg(t) (p, x)2+k < ∞.
x∈M
Proof. Note first that on the cone we can choose a δ > 0 small enough such that for every
x0 ∈ C \ {o} we have
1
|Rm|(x, 0) ≤
(2δr(x0 ))2
for any x ∈ BgC (x0 , 2δr(x0 )) and
Volg(0) (B(x0 , 2δr(x0 ))) ≥ (1 − /2)ωn (2δr(x0 ))n
where is the universal constant in the pseudolocality theorem in [Per02].
Now pick any point x0 in M \ K. From the convergence of g(t) it follows that we can find s0 = s(x0 ) > 0
small enough such that
1
|Rm|(x, s) ≤
(δr(x0 ))2
for x ∈ Bg(s) (x0 , δr(x0 )), all s ≤ s0 and
Volg(s) (B(x0 , δr(x0 ))) ≥ (1 − )ωn (δr(x0 ))n
28
3. RICCI FLOWS ATTAINING THE CONE AS AN INITIAL CONDITION
for all s ≤ s0 .
By the Pseudolocality theorem it follows that
1
(δr(x0 ))2
|Rm|(x, s0 + t) ≤
for x ∈ Bg(t+s0 ) (x0 , δr(x0 )) and t ∈ [0, min{0, T − s0 , (δr(x0 ))2 }]. Choosing s0 smaller if necessary we
may assume that T − s0 > 0. In particular
1
|Rm|(x0 , s0 + t) ≤
(δr(x0 ))2
for t ∈ [0, T − s0 ] if r(x0 )2 ≥ T /(δ)2 . After sending s0 to 0 it follows that
√
|Rm|(x, t) ≤ C/r(x)2
for t ∈ [0, T 0 ] if r(x) ≥ T 0 /(δ), T 0 ≤ T . The estimates for the higher derivatives of the curvature
follow from lemma A.4 in [Top10].
3.3. Ricci flows coming out of Ricci flat cones
Now we assume that the starting cone is Ricci flat. Of course in this case we expect the solution of the
Ricci flow to stay close to the original cone. The intuition comes from mean curvature flow starting
with stationary hypercones which are not area-minimizing cones. In the context of mean curvature flow
this is the criterion for the existence of a non-trivial forward evolution. For the Ricci flow it is not
clear yet what the appropriate criterion should be. However, if such a flow exists then the difference of
the evolving and the stationary metric is exponentially decaying to zero as x → ∞. We show this by
construction of a comparison function on ΩT := ∪t∈[0,T ] (M \ Kt × {t}) where Kt is a compact set whose
√
diameter grows proportionally to t.
Theorem 3.3.1. Let (M, g(t)) be a complete Ricci flow with bounded curvature starting from the Ricci
flat cone (C, gC ). Then for each κ > 0 there exist constants C = C(κ) and λ = λ(κ) such that for
(x, t) ∈ M \ K × (0, T ] with r(x)2 /4t > λ
κ+1
|Rc|(x, t) ≤ C σ(x,t)
t
e−σ(x,t) ,
where σ(x, t) = r(x)2 /4t.
We can compare this result with concrete solutions for the Ricci flow starting from a Ricci flat cone.
Example 3.3.2 (2-dimensional expanding Ricci solitons). The two-dimensional gradient expanding Ricci
soliton coming out of a flat cone mentioned in the introduction has indeed the required curvature decay.
Following [CLN06] it can be shown that the Gauss curvature of this Ricci soliton1 is
1
1
1
Kg (x, t) =
−
,
t F (r) 2
where
2
F (r) =
,
2
2
2
W
−
1
exp
−
1
−
r
+
1
F (0)
F (0)
W being the Lambert function2. One can check that for x ∈ R2 and t > 0
√ Z r(x)
|x| = t
F (r)dr
0
√
and so for large r we have |x| ∼ 2 tr (where |x| is the distance from the origin in the cone metric).
Studying the asymptotics of W near 0 it follows that Kg (x, t) ∼ t−1 exp(−|x|2 /4t). These expanding
Ricci solitons have negative scalar curvature if F (0) > 2 and positive scalar curvature if F (0) ∈ (0, 2).
If F (0) = 2 then F (r) = 2 for all r and the resulting expanding Ricci soliton is the Gaussian soliton on
R2 .
1g(t) = t(F (r)2 dr 2 + r 2 dθ 2 ), θ ∈ R/F (0)2π and R2 is parametrized as (0, ∞) × S 1 (F (0)).
2W (x) exp(W (x)) = x
3.3. RICCI FLOWS COMING OUT OF RICCI FLAT CONES
29
Another example that was already described before is the following
Example 3.3.3. The expanding Kähler Ricci solitons constructed in [FIK03] are Ricci flows coming
out of Ricci flat Kähler cones. These expanding Ricci solitons are rotationally symmetric and so the
expander equation can be reduced to an ODE √
at t = 1/4 for the potential P of the expander metric.
¯ where P = P (log(|z|2 )) is a smooth function
More precisely, the solution ω is of the form −1∂ ∂P
√
n
defined on C /Zk \ {0} (k > n ,k ∈ N), where Zk acts diagonally on Cn sending ~z to exp(2π −1j/k)~z
for j = 0, ..., k − 1. An appropriate choice of the free parameters gives a complete metric by gluing in a
CP n−1 at 0. For the Ricci flat cone the ODE for P is
Prr =
−an ea e−Pr
+ Pr
Prn−1
where a = k − n > 0 and r = log(|z|2 ). Setting φ := Pr one can check that the eigenvalues of the
resulting Ricci curvature Rc with respect to the expander metric g are
λn = an ea e−φ φ−n (φ + (n − 1))
and
−an ea e−φ
.
φn
Finally it is shown in [FIK03] that φ behaves like er = |z|2 near infinity and so it follows that
λ1 = ... = λn−1 =
|Rc|(z, 1/4) = O(exp(−|z|2 ))
on Cn /Zk \ {0} as z → ∞. And for arbitrary t > 0
|Rc|(z, t) = O(exp(−|z|2 /4t))
as z → ∞.
Remark 3.3.4. Note that in the example above the scalar curvature is positive (unlike the Ricci curvature). In fact
an ea e−φ
Rg =
.
φn−1
Before we start with the proof we mention one more example from the world of mean curvature flow.
As explained in the introduction there exist self-similar solutions of the mean curvature flow coming out
of the unstable Simons cones. And these solutions are exponentially asymptotic to the original cone as
shown in [Ilm95].
Example 3.3.5. In [Ilm95] expanders with a SO(p) × SO(q) symmetry coming out of the unstable
Simons cones Cp,q are constructed. These solutions are described as Graphs over Rp :
M1 = {x = (y, z) ∈ Rp × Rq : |z| = u(|y|)}
where the mean curvature H satisfies
hx, ~ν i
= 0,
2
ν being the unit normal vector. It follows from the calculations in [Ilm95] that near infinity the mean
curvature behaves like
2
H(x) ≈ C1 exp − |x|4 .
H−
Remark 3.3.6. This example should also be compared with the expanders in [JLT10]. In [LN12] it is
shown that these expanders approach the original cone exponentially fast.
Proof of the theorem.
3
First recall the evolution equation for the Ricci curvature (see [CLN06])
∂
Rc(g(t)) = ∆L Rc(g(t)).
∂t
3Throughout the proof will write g instead of ϕ∗ g.
30
3. RICCI FLOWS ATTAINING THE CONE AS AN INITIAL CONDITION
where ∆L is the Lichnerowicz Laplacian acting on symmetric 2-tensors with respect to the evolving
metric
(∆L h)ij = (∆h)ij + 2Rmikjl hkl − Rcik hkj − Rckj hik .
In particular
∂
|Rc|2 = 2h∆Rc, Rci + 4hRm ∗ Rc, Rci
∂t
≤ ∆|Rc|2 − 2|∇Rc|2 + 4|Rm||Rc|2 .
Then for any > 0
∆|Rc|2 + 4|Rm|||Rc|2 − 2|∇Rc|2 + 2t2
∂p2 2
p
t + |Rc|2 ≤
∂t
2 2 t2 + |Rc|2
where
p
t2 2 + |Rc|2 is smooth in M × (0, T ] and continuous on M \ K × [0, T ]. And so
p
p
∂p 2 2
t2
t + |Rc|2 − ∆ 2 t2 + |Rc|2 ≤ 2|Rm| t2 2 + |Rc|2 + p
.
∂t
t2 2 + |Rc|2
Indeed
∆
p
∆|Rc|2
|∇|Rc|2 |2
2 t2 + |Rc|2 = p
− p
3
2 2 t2 + |Rc|2
4 2 t2 + |Rc|2
∆|Rc|2
|∇Rc|2 |Rc|2
≥ p
−p
3
2
2
2
2 t + |Rc|
2 t2 + |Rc|2
∆|Rc|2
≥ p
−
2 2 t2 + |Rc|2
!
|∇Rc|2
p
2 t2 + |Rc|2
|Rc|2
2 t2 + |Rc|2
∆|Rc|2
|∇Rc|2
≥ p
−p
.
2 2 t2 + |Rc|2
2 t2 + |Rc|2
Let σR (x, t) := (r(x) − R)2 /4t, x ∈ M \ K, and for λ > 0 define
Ωλ := {(x, t) ∈ M \ K × [0, T ) : σR (x, t) > λ, r(x) > R}.
Recall that on Ωλ (for λ sufficiently large) we can bound the curvature by
C
|Rm|(x, t) ≤
.
4λt + R2
where C does not depend on t. Let η := C/λ < 1 (i.e. λ sufficiently large), then for
p
t
|Rc|2 + 2 t2 −
u := (4λt + R2 )−η
1−η
we have
∂
u − ∆u ≤ (4λt + R2 )−η
∂t
4λη
2|Rm| −
4λt + R2
p
|Rc|2 + 2 t2
!!
t2
4ληt
−
+ p
+
1 − η (4λt + R2 )(1 − η)
2 t2 + |Rc|2
p
2C
4C
−
|Rc|2 + 2 t2
≤ (4λt + R2 )−η
2
2
4λt + R
4λt + R
η
+ −
+
1−η 1−η
≤ 0.
Now define
v(x, t) :=
1
σ µ e−σR ,
(4λt + R2 )1+η R
3.3. RICCI FLOWS COMING OUT OF RICCI FLAT CONES
31
with µ > 0 to be determined later. Further note that (using g(0) = gC )
n
(n − 1)R
(r − R)2
0
00
2
0
∆g(0) f (σR ) = f (σR )∆g(0) σR + f (σR )|∇σR |g(0) = f (σR )
.
−
+ f 00 (σR )
2t
2rt
4t2
µ
So if f (σR ) = σR
exp(−σR ) then
µ
2µ
n
(n − 1)R
µ(µ − 1)
σR
∆g(0) f (σR ) =
−
−1
−
+
+1
f (σR )
2
σR
2t
2rt
σR
σR
t
and
µ σ ∂
∂σR
R
f (σR ) = f 0 (σR )
= f (σR ) − +
∂t
∂t
t
t
Then (note that g(x, t) is a smooth metric in x and t on each Ωλ ∩ M × {t})
∂
∂
− ∆g(t) v =
− ∆g(0) v + (∆g(0) − ∆g(t) )v
∂t
∂t
|
{z
}
=:w
µ − (1 + η)(4λt)/(4λt + R2 ) + (n/2) − R(n − 1)/(2r)
=
t
µ(µ − 1) + µ(n/2) − µR(n − 1)/r
−
v+w
tσR
So if µ = η + 1 + γ for some γ > 0, then on Ωλ
γ
µ(µ − 1) + µ(n/2) − R(n − 1)/r
∂
− ∆g(t) v ≥
−
v + w.
∂t
t
tλ
And if λ (independent of R) is large enough
γ
∂
− ∆g(t) v ≥
v + w.
∂t
2t
Next we can estimate w:
Proposition 3.3.7. On Ωλ the function w can be estimated by
C1
−1
|w| ≤
+ C2 σR v
t
where C1 and C2 depend only on T and a lower bound for λ.
Proof. Note first that
−1
−2
−1
∆g(t) v = ((µσR
− 1)2 − µσR
)|∇σR |2g(t) + (µσR
− 1)∆g(t) σR )v
−2
−1
−1
= (µ(µ − 1)σR
− 2µσR
+ 1)|∇σR |2g(t) + (µσR
− 1)∆g(t) σR )v
Hence it remains to investigate |∇σR |2g(t) − |∇σR |2g(0) and ∆g(t) σR − ∆g(0) σR . So let (x, t) ∈ Ωλ
C
d
|∇σR (x, t)|2g(s) ≤ 2|Rc(x, s)|g(s) |∇σR (x, t)|2g(s) ≤
|∇σR (x, t)|2g(s) .
ds
r(x)2
It follows that
C
.
t
Indeed for some constant C1 depending only on a lower bound for λ
|∇σR |2g(t) − |∇σR |2g(0) ≤
Ct
|∇σR |2g(t) − |∇σR |2g(0) ≤ (e r(x)2 − 1)|∇σR |2g(0) ≤
C1 σR
.
σR t
(3.1)
32
3. RICCI FLOWS ATTAINING THE CONE AS AN INITIAL CONDITION
Similarly we can find constants C2 , C3 , and C4 depending only on a lower bound for λ
n
(∆g(t) − ∆g(0) )σR (x, t) ≤ |g(t)−1 − g(0)−1 |g(0) + |Γ(x, t) − Γ(x, 0)|g(0) |∇σR (x, t)|g(0)
2t
C3
C4 t (r(x) − R)
≤
+
2
r(x)
r(x)3
t
C2
≤
.
tσR (x, t)
(3.2)
So
−1
|w| ≤ (C1 /t + C2 σR
)v.
In particular C1 and C2 do not depend on µ.
(3.3)
So if we choose λ and γ large enough (again independent of R) we get
∂
− ∆g(t) v ≥ 0
∂t
on Ωλ .
Lemma 3.3.8. For sufficiently large λ there exists a uniform Cλ such that
1+γ+η −σR
|Rc|(x, t) ≤ Cλ (4λt + R2 )η v = Cλ (4λt + R2 )−1 σR
e
Proof. Note first that
h(x, t) := Cλ v(x, t) − u (x, t) ≥ 0,
on {0} × {r(x) > R} ∪ {(x, t) : t ∈ (0, T ], σR = λ}
where
Cλ = Cλ−1−γ−η eλ .
and
∂h
− ∆g(t) h ≥ 0, (x, t) ∈ Ωλ .
∂t
We claim that h ≥ 0 in Ωλ . If the infimum is attained in the interior of Ωλ then it follows by the
maximum principle that h ≥ 0. Now suppose that inf h < 0 and let (xj , tj ) be a sequence in Ωλ such
that limj→∞ h(xj , tj ) = inf Ωλ h. By the observations above it follows that tj → 0 and r(xj ) → R.
But since u is continuous in Ωλ we have limj→∞ u (xj , tj ) = 0 and limj→∞ v(xj , tj ) ≥ 0. This is a
contradiction.
Letting R → 0 and → 0 we get
|Rc|(x, t) ≤ C̃λ t−1 σ 1+γ+η e−σ ,
and so after integrating
|g(x, t) − g(x, 0)|gC = O(σ µ−1 )e−σ .
We can repeat the process above. Namely, from the results above we have an even better bound for w,
i.e. |w| tends to 0 as λ increases. More precisely, the right hand side of (3.1) can be improved such that
µ
it is of the form (C 0 /t)σR
exp(−σR ). So for any γ > 0 we can always find a sufficiently large λ such that
γ
µ(µ − 1) + µ(n/2) − R(n − 1)/r
γ
−
> ,
t
tλ
2t
and such that on Ωλ we have |w| ≤
γ
2t .
Hence for such a λ
∂v
− ∆v ≥ 0.
∂t
And as before we can conclude that |Rc| ≤ Cλ v. Obviously we can choose γ > 0 and η > 0 as small as
we want and hence
κ+1
|Rc| = O(σR
/te−σR ),
for every κ > 0 and so
κ −σR
|g(x, t) − g(x, 0)|gC = O(σR
e
)
3.3. RICCI FLOWS COMING OUT OF RICCI FLAT CONES
33
for every κ > 0. Note however, that we cannot let κ → 0 since the constant factor definitely depends
on κ (but not on R). And again, letting R → 0, we have
κ+1
|Rc| = O σ t exp(−σ)
and
|g(x, t) − g(x, 0)|gC = O(σ κ e−σ ).
As a corollary we can conclude that the Kähler class of any Kähler Ricci flow coming from a cone must
be proportional to c1 (M ). In particular [ω(t)] ∈ Hc2 (M ) for all t (this also follows from proposition 2.8.3
in the previous chapter).
Corollary 3.3.9. Assume that (M, J, g(t)) is a Kähler Ricci flow emerging from a (Ricci flat) Kähler
cone (as defined above). Then [ω(1)] = −c1 (M ) and [ω(1)] ∈ Hc2 (M ).
Proof. Take a compact 2-cycle Σ ∈ H2 (M ). Then
Z
Z
Z
Z
Z
0 = lim [ω(t)] = [ω(1)] − lim (1 − t)
2πc1 (M ) = [ω(1)] −
c1 (M ).
t→0
Σ
Σ
t→0
Σ
Σ
Σ
And if Σ is in M \ K then we can find [Σ̃] ∈ H2 (S) such [ϕ({1} × Σ̃)] = [Σ]. So by Stokes’ theorem and
the fact proven above that ϕ∗ ρ decays exponentially compared to ωC it follows that
Z
Z
Z
[ω(1)] =
[ϕ∗ ω(1)] = lim
[ϕ∗ ρ] = 0,
Σ
i.e. ω(1) ∈
Hc1,1 (M, R)
{1}×Σ̃
r→∞
{r}×Σ
CHAPTER 4
Expanding Ricci Solitons
4.1. Introduction
A solution g(t) of the Ricci flow
∂g
= −2Rc(g(t))
∂t
is called an expanding gradient Ricci soliton if it is of the form
g(t) = tϕ∗t g(1),
where ϕt is a family of diffeomorphisms generated by a (gradient) vector field X (= −∇g f /t) with
ϕ1 = id. It follows that the metrics g(t), t > 0, solve the elliptic partial differential equation (note that
f depends on t)
g(t)
Rc(g(t)) − ∇2 f +
= 0.
(4.1)
2t
As such, expanding Ricci solitons provide local models for Ricci flows coming out of singularities. They
also appear as blow down limits of eternal solutions of the Ricci flow. In this chapter we study the
behavior of expanding Ricci solitons near the singular time t = 0 and show that under the assumption
of quadratic curvature decay the flow converges to an asymptotic cone as t & 0. In particular, expanding
Ricci solitons are the prototypes of Ricci flows that smooth out conical singularities. We should point
out that an important part of the proof (the Gromov Hausdorff convergence) has already been provided
in [CD11]. Expanding Ricci solitons show up in the following to situations.
4.1.1. Ricci flows with singular initial conditions. Let us assume that (M, g(t))t∈[0,T ] is a, yet
to be defined, weak Ricci flow with a singular initial condition such that the flow is smooth for t ∈ (0, T ]
, λj & 0,
and such that supt∈(0,T ] tkRmk∞ < ∞. Then, ideally, the sequence (M, xj , λ−1
j g(λj t))t∈(0,λ−1
j T]
subconverges to an expanding Ricci soliton that flows out of the tangent cone with limj→∞ xj as its tip.
Hence expanding Ricci solitons model the behavior of the Ricci flow after a singularity.
4.1.2. Blow down limits of the Ricci flow. A Ricci flow (M, g(t))t∈[0,∞) is called a type III
solution if supt∈[0,∞) tkRmt k∞ < ∞. It is expected that the sequence (M, λj g(λ−1
j t)), λj & 0, subconverges to an expanding Ricci soliton. This has been proven under various positivity assumptions. Cao
showed in [Cao97] that for a non-compact type III solution for the Kähler Ricci flow with non-negative
bisectional curvature and positive Ricci curvature such that the maximum of tR(x, t) is assumed in space
time the sequence (M, x, t−1 g(t)) converges to an expanding Kähler Ricci soliton. Schulze and Simon
show in [SS10] that the same is true for type III Ricci flows with non-negative curvature operator and
positive asymptotic volume ratio giving an expanding Ricci soliton coming out of the unique asymptotic
cone of M . Note that without the assumption of nonnegative sectional curvature the asymptotic cone
(if it exists) is usually not unique (for a given manifold (M, g)).
4.2. Basic properties of expanding Ricci solitons
Taking the divergence of equation (4.1) it follows that1
R(g(t)) + |∇f |2 −
f
= C1 (t),
t
1We denote by R(g) and R the scalar curvature with respect to g.
g
35
(4.2)
36
4. EXPANDING RICCI SOLITONS
for some constant C1 (t) depending only on t. More precisely, the constant C1 (t) is of the form C1 /t,
C1 ∈ R. Since
∂
R(g(t))
∂t
∂
R(g(t))
∂t
=
∆g(t) R(g(t)) + 2|Rc(g(t))|2g(t) ≥ ∆g(t) R(g(t)) +
=
∂ R(g) ◦ ϕt
1
= X(R(g(t))) − R(g(t))
∂t
t
t
2
R(g(t))2
n
we also have
2
R(g(t))
R(g(t))2 +
.
n
t
And so if the maximum principle holds for (M, g) (if (M, g) has bounded curvature) it follows that
n
and hence ∆f ≥ 0 and f is bounded from below for t > 0.
inf M R(g(t)) ≥ − 2t
0 ≥ ∆R(g(t)) − X(R(g(t))) +
Lemma 4.2.1. Let (M, f, g) be an expanding gradient Ricci soliton and suppose that
sup |Rm|(x)d(p, x)2 =: A0 (g) < ∞
x∈M
and let
c0 :=
|Rm|(x).
sup
x∈B(p,1)
then
2
1
4 d(p, x)
+ |∇f |(p)d(p, x) + π(A0 (g) + c0 )(n − 1)d(p, x)
≥ f (x) − f (p) ≥ 41 d(p, x)2 − |∇f |(p)d(p, x) − π(A0 (g) + c0 )(n − 1)d(p, x).
Proof. Let γ(t) be a geodesic joining p and x. Then
d2
˙ + 1.
f (γ(t)) = ∇2 f (γ(t))(γ̇(t), γ̇(t)) = Rc(γ(t))(γ̇(t), γ(t))
2
dt
2
Now integrating by parts gives
Z d(p,x)
d2
(d(p, x) − τ ) 2 f (γ(tτ ))dτ = d(p, x)h∇f (p), γ̇(0)i + f (x) − f (p),
dτ
0
while
Z
0
d(p,x)
d2
(d(p, x) − τ ) 2 f (γ(τ ))dτ =
dτ
Z
d(p,x)
(d(p, x) − τ )Rc(γ(τ ))(γ̇(τ ), γ̇(τ )) +
0
Z
d(p,x)
(d(p, x) − τ )
dτ
2
2(n − 1)(c0 + A0 (g))(d(p, x) − τ )
dτ
1 + τ2
≥
d(p,x)2
4
−
≥
d(p,x)2
4
− 2 arctan(d(p, x))(2(n − 1)(c0 + A0 (g)))d(p, x)
≥
2
d(p,x)
4
0
− π(2(n − 1)(c0 + A0 (g)))d(p, x)
which proves the second inequality in the lemma. The first one works similarly.
It follows that, under the assumptions above, f attains its minimum in M . And if we take p to be that
point, i.e. f (p) = inf x∈M f (x) then ∇f (p) = 0 and hence f (p) = C1 − Rg (p). Without loss of generality
we can assume (after adding a constant to f ) that C1 = Rg (p) and so inf x∈M f (x) = f (p) = 0. Moreover
2
1
4 d(p, x)
+ c1 d(p, x) ≥ f (x) ≥ 41 d(p, x)2 − c1 d(p, x),
where
c1 := π(2(n − 1)(c0 + A0 (g)))
4.3. CONVERGENCE OF RICCI SOLITONS
37
4.3. Convergence of Ricci solitons
Before we start a word on notation: let g(x, t) be a time dependent metric on M and t ∈ (0, T ]. When
we write |∇k Rm|(x, t) we mean the k-th covariant derivative with respect to the metric g(x, t) of the
Riemannian curvature of g(x, t) at x and t. On the other hand, if we write |∇k Q(x, t)|g(x) for some time
dependent tensor field Q(x, t) on M , t ∈ [0, T ] then we mean the k-th covariant derivative of Q with
respect to g(x) measured in g(x).
Suppose that (M, f, g) is an expanding gradient Ricci soliton. Then using the observations of the
previous section one can prove the following counterpart to the theorem 3.2.3 we proved earlier:
Theorem 4.3.1. Let (M, f, g) be a non-compact expanding gradient Ricci soliton such that for some
point p ∈ M and all k ∈ N0
Ak (g) := sup |∇k Rm|(x)d(p, x)2+k < ∞
x∈M
then the associated Ricci flow (M, g(t)) converges to a Riemannian cone C as t → 0.
Remark 4.3.2. A version of this theorem has already been proven in [CD11]. In particular we use
their convergence result in the Gromov Hausdorff topology and merely spell out the details of remark
1.5 in [CD11]. In particular, we show that the cone is unique and the convergence is actually smooth.
Proof. We already know from [CD11] that (M, g(t)) converges subsequentially in C 1,α to a cone
(away form the vertex) and in the Gromov Hausdorff sense as a metric space if we have quadratic
curvature decay. And the proof can be modified to show that (M, g(t)) converges subsequentially
smoothly (away from the vertex) under the curvature assumptions above. It remains to show that the
limit is unique, i.e. that (M, g(t)) converges to the same cone independent of the choice of subsequence.
Unless we have non-negative sectional curvature we cannot assume in general that the tangent cone at
infinity of M is unique2. So let
Mc
:= {x ∈ M : lim inf tf (x, t) > c},
M0
:= {x ∈ M : lim inf tf (x, t) > 0},
t>0
t>0
where f (x, t) is defined as
∂
f (x, t) =
∂t
f (x, 1) =
−|∇f |2g(t) (x, t)
f (x).
Note that f (x, t) = f (ϕt (x), 1) where ϕt is the one parameter group of diffeomorphisms
∇f (ϕt (x))
d
ϕt (x) = −
,
dt
t
and ∇f is the gradient of f = f (1) with respect to g = g(1), i.e. the metric at t = 1.
Lemma 4.3.3. Let (M, g(t)) be as above. Then there exists a constant B(0, c) such that on Mc
|Rm|(x, t) ≤ B(0, c),
for c > 0 independent of t.
Proof of the lemma. Note first that
|Rm|(x, t) = t−1 |Rm|(ϕt (x)),
By the observations above
|∇f |2g (ϕt (x))
∂
f (ϕt (x)) = −
,
∂t
t
2of course it follows from the theorem that the tangent cone exists and is unique.
38
4. EXPANDING RICCI SOLITONS
and so
∂
tf (ϕt (x)) = (f − |∇f |2g )(ϕt (x)) = Rg (ϕt (x))
∂t
and integrating gives
Z
tf (ϕt (x)) = sf (ϕs (x)) +
t
Rg (ϕτ (x)) dτ.
s
Moreover, for x ∈ Mc ,
tf (ϕt (x)) ≥ lim inf sf (ϕs (x)) − t sup |Rg (y)| > c − t sup |Rg (y)|
s→0
y∈M
i.e
y∈M
c
− sup |Rg (y)|.
t y∈M
f (ϕt (x)) >
And so, letting r(x) := d(p, x)
1
c
r(ϕt (x))2 + c1 r(ϕt (x))) ≥ − sup |Rg (y)|.
4
t y∈M
If
t<
c
supy∈M |Rg (y)| + c21
we have
r(ϕt (x)) ≥
1
2c1
r
c
− sup |Rg (y)|.
t y∈M
This means that for x ∈ Mc ,
|Rm|(x, t) = t−1 |Rm|(ϕt (x)) ≤ sup |Rm|(y)t−1
0 +
y∈M
where
t0 :=
8A0 (g)c21
C(g)
≤
=: B(0, c),
c
c
c
,
2(supy∈M |Rg (y)| + c21 )
and C(g) is a constant that depends only on the metric and its derivatives.
Lemma 4.3.4. Let (M, g(t)) be as above. Then on Mc we have
sup |∇k Rm|(x, t) ≤ B(k, c)
M
for some uniform constant B(k, c) depending only on k and c.
Proof. Arguing as above we have
−1−k/2
|∇k Rm|(x, t) = t−1−k/2 |∇k Rm|(ϕt (x)) ≤ sup |∇k Rm|(y)t0
y∈M
+
8Ak (g)c21
=: B(k, c).
c
Lemma 4.3.5. Let f (x, t) and (M, g) be as above. Then tf (x, t) converges uniformly in C 0 (M ) and
locally in C ∞ (M0 ) with respect to g to a smooth function f0 on M0 . Moreover ∇f0 (x) 6= 0 for all
x ∈ M0 .
Proof. By the proof of the previous lemma we have
Z t
tf (x, t) − sf (x, s) =
Rg (ϕτ (x))dτ
s
0
which proves that tf (x, t) is a Cauchy sequence in C (M ) and hence converges uniformly as t → 0.
Now we compare g and g(t) on M0 . First let x ∈ M0 i.e. limt→0 tf (x, t) > c for some c > 0. Then for
0<t<1
C(0, c)g(x) = exp (2(n − 1)B(0, c)) g(x) ≥ g(x, t) ≥ exp (−2(n − 1)B(0, c)) g(x) = C(0, c)−1 g(x)
4.3. CONVERGENCE OF RICCI SOLITONS
39
and by induction (see lemma below) it can be shown that for each k ≥ 1 there exists a constant C(k, c)
such that
|∇k g(x, t)|g(x) ≤ C(k, c).
and a constant L(k, c) such that
|∇k Rm(x, t)|g(x) ≤ L(k, c)
So again by induction (see lemma below) it follows that for each k ≥ 2 there exists a constant K(k, c)
such that
|∇k tf (x, t)|g ≤ K(k, c).
∞
(M0 ). And hence f0 (x) = limt→0 tf (x, t) is a smooth function
It follows that tf (x, t) converges in Cloc
on M0 . Finally the claim that |∇f0 (x)|g 6= 0 for x ∈ M0 follows from equation (4.2).
Lemma 4.3.6. For each k ≥ 0 there exist constants C(k, c) and L(k, c) such that for t ∈ (0, 1]
|∇k g(x, t)|g(x) ≤ C(k, c)
and
|∇k Rm(x, t)|g(x) ≤ L(k, c).
Proof. We prove this lemma by induction. The case k = 0 has already been treated above so
˜ the Levi Civita connection with respect
assume that the lemma is true for all l < k. We denote by ∇
to g(x, t). Then
∂ k
|∇ g(x, t)|g(x) ≤ 2|∇k Rc(x, t)|g(x)
∂t
˜ k )Rc(x, t)|g(x) + C(0, c)k/2+1 |∇Rc|(x,
˜
≤ 2 |(∇k − ∇
t)


k−1
X
˜
≤ 2
|∇k−j g(x, t)|g(x) |∇j Rc(x, t)|g(x) + C(0, c)k/2+1 |∇Rc|(x,
t)
j=0

k−1
X
≤ 2

C(k − j, c)L(j, c) + C(0, c)k/2+1 B(k, c) + 2|∇k g(x, t)|g(x) L(0, c).
j=0
Integrating implies that
|∇k g(x, t)|g(x) ≤ C(k, c),
for some constant C(k, c). Similarly
˜ k )Rm(x, t)|g(x) + C(0, c)k/2+2 |∇
˜ k Rm|(x, t) ≤ L(k, c).
|∇k Rm(x, t)|g(x) ≤ |(∇k − ∇
We continue with the proof of the theorem. By the observations above we have for s < t ≤ 1
Z t
k(Mc , g(t)) − (Mc , g(s))k(C k (Mc ),g(t)) ≤ Rc(x,
τ
)
dτ
k
s
≤
k
X
(C (Mc ),g)
Z
sup
j=0 x∈Mc
≤ |t − s|
k
X
t
|∇j Rc(x, τ )|g dτ
s
L(j, c)
j=0
and it follows that g(t) converges uniformly in C k (Mc ) for every k ≥ 0 and c > 0 to a unique metric
g(0) on M0 . Let f0 (x) := limt→0 tf (x, t) which is a smooth function on M0 with ∇f0 (x) 6= 0 for all
x ∈ M0 (by the observations above) and let L := {x ∈ M0 : f0 (x) = 1/4}. Now note that (g0 := g(0)
on M0 ) taking the limit in (4.1) it follows that
∇2 f0 (x) =
g0 (x)
2
40
4. EXPANDING RICCI SOLITONS
on M0 where ∇2 f0 is the Hessian with respect to g0 . Set
g̃(x) := (4f0 (x))−1 g0 (x)
and
f˜(x) := log(4f0 (x))
for x ∈ M0 . Then (M0 , g̃) is a complete manifold and
˜ 2 f˜(x) = 0
∇
and so by the de Rham splitting theorem M0 is isometric to (R × L̃, ds2 + g̃L̃ ) where (L̃, g̃L̃ ) = f˜−1 (0).
It follows that (M0 , g0 ) is isometric to a cone (R>0 × L, dr2 + r2 gL ) where gL = ı∗L g0 . Finally M \ M0
is a compact (non-empty since p ∈ M \ M0 ) set in the original topology since f0−1 (0) is a closed subset
of a bounded set:
{x ∈ M : f0 (x) = 0} ⊂ {x ∈ M : f (x) ≤ sup |Rg (x)|}.
x∈M
CHAPTER 5
Existence of Expanding Kähler Ricci Solitons
5.1. Introduction
As explained in the previous chapter we are interested in self similar solutions of the Ricci flow in order
to understand its behavior near singularities. These singularities are particularly well understood in
the Kähler case. In an ideal case the Kähler Ricci flow develops a singularity along a subvariety (if it
does not collapse to a lower dimensional variety) as t approaches the singular time and converges in the
Gromov Hausdorff sense to a singular space and smoothly away from the subvariety. The limit should
be a possibly singular variety. To restart the flow on a singular variety it is necessary to understand
the Kähler Ricci flow with singular initial conditions. A particularly simple example of a singularity
is a Ricci flat Kähler cone (which, including the vertex, is itself an affine algebraic variety). Here we
construct solutions of the Ricci flow that come out of such cones and smooth out the singularity by
adding in several subvarieties (resolving the singularity). In particular these solutions are topologically
different for positive times but birationally equivalent to the original cone.
Tian and Song laid out a conjectural program in [ST09] to show that on a compact algebraic variety the
Kähler Ricci flow does not have to be stopped before the singularity (as in the Perelman program) but
can actually be extended to flow through a singularity performing algebraic surgeries such as divisorial
contractions or flips. In other words, one gets a sequence of varieties X1 , X2 , X3 , ... and times T1 < T2 <
T3 .... such that Xj is related to Xj+1 by one of the algebraic surgeries mentioned above and Kähler Ricci
flows gj (t) on Xj × (Tj−1 , Tj ) such that limt→Tj+1 (Xj , gj (t)) = limt→Tj+1 (Xj+1 , gj+1 (t)) in a suitable
sense.
This program allows the varieties to have log terminal singularities (not only at the singular times where
the singularities can be much more complicated). A typical example of such a singularity is a quotient
of Cn by a finite subgroup of U (n). In dimension 2 log terminal singularities will always be of this form.
The Ricci flows constructed here (self-similar exapnding solutions) will smooth out such singularities
therefore creating an ambiguity in the continuation of the Ricci flow (see remark 5.3.9).
We begin by recalling some well known facts about Kähler Ricci solitons. In the following section
we reduce the soliton equation to a Monge Ampere equation and prove that a unique solution of this
equation exists on resolutions M of Ricci flat Kähler cones C with KM > 0. Finally we list a number
of old and new examples to which this theorem applies.
5.2. Kähler Ricci solitons
In this section we √
discuss some general properties of Kähler Ricci solitons. Let
√ (M, J, g) be a Kähler
manifold, h := g − −1ωg its associated hermitian metric, and R(h) := Rc(g)− −1ρ(ωg ) the hermitian
Ricci tensor. (M, J, g) is a Ricci soliton as a real Riemannian manifold if for some vector field ζ on M
− Lζ g + λg = −Rc(g),
(5.1)
for λ ∈ {−1, 0, 1}. We say that (M, J, g) is a Kähler Ricci soliton if in addition
− Lζ ωg + λωg = −ρ(ωg ).
(5.2)
This makes (M, J, g) a hermitian soliton in the sense that
− Lζ h + λh = −R(h).
(5.3)
Since g = ω( · , J·) subtracting (5.1) from (5.2) leaves Lζ J = 0 and so it follows that ζ has to be a
holomorphic vector field. Equation (5.1) alone does not imply that ζ is holomorphic, however if ζ is a
41
42
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
gradient then ζ has to be holomorphic (see [FIK03]). Conversely, if (M, J, g) is a Kähler Ricci soliton
then LJζ g = 0. Indeed,
LJζ gij = ∇i (Jζ)j + ∇j (Jζ)i
= J(∇i ζ)j + J(∇j ζ)i
= Jjk (Rcik + λgik − (∇k ζ)i ) + Jik (Rcjk + λgjk − (∇k ζ)j )
= −Jjk (∇k ζ)i − Jik (∇k ζ)j
= −Jjk (∇i ζ)k − Jik (∇j ζ)k
= −∇i (Jζ)j − ∇j (Jζ)i
= −LJζ gij ,
where we use equation 5.1 in line 3, the compatibility of g and Rc in line 4, and Lζ J = 0 in line 5. Then
by proposition
2.1.7 it follows that ζ is a gradient if M is simply connected or H 0,1 (M ) = 0. Now let
√
¯ = 0. So (h, X) satisfies
X := ζ − −1Jζ. Then X is a section on T M 1,0 and ∇0,1 X = ∂X
−LX h + λh = −R(h).
and in particular
−LX ωg + λωg = −ρ(ωg ).
¯ := ∇0,1 unless otherwise indicated. Furthermore z1 , ..., zn
From now on we will write ∇ := ∇ and ∇
will be local holomorphic coordinates on M . By holomorphic vector field we always refer to the (1, 0)part of a real holomorphic vector field. In these coordinates we have
√
ωg = −1gi̄ dzi ∧ dz̄̄
1,0
and
X = X j (z)
∂
∂zj
j
where
√ X (z) are holomorphic functions. We will write ω instead of ωg if it is clear what g is and ωϕ for
¯
ω + −1∂ ∂ϕ.
Remark 5.2.1. The geometry of Kähler Ricci solitons is described in [Bry08]. In particular, the
following is observed: If X is a complex holomorphic vector field on M such that X = (∇f )1,0 for some
real valued function f then the singular set Z of X (the points where X vanishes) is a complex subvariety
of M . Moreover, since Z is also the zero locus of the Killing vector field =(X), the imaginary part of
X, it follows that Z is totally geodesic (theorem 5.5 on page 60 in [Kob72]) and therefore nonsingular.
Hence Z is a disjoint union of smooth complex subvarieties.
5.2.1. Necessary conditions. Let (M, J, g) be a Kähler Ricci soliton and let Σ be a compact
complex curve in M and [Σ] its associated 2-cycle. Recall that ρ(ωg ) is a representative of 2πc1 (M ). So
integrating equation (5.2) over Σ gives
Z
−LX ωg + λVolg (Σ) = −2πhc1 (M ), [Σ]i.
Σ
Since ωg is closed LX ωg is exact and so
λVol(C) = −2πhc1 (M ), [C]i.
In particular −KM (KM ) is numerically positive if λ = −1 (λ = 1). Moreover, it follows from √
equation
¯
¯
(5.2) that λ[ωg ] = −2πc1 (M ). If M is compact the ∂ ∂-Lemma
implies that λωg = −2πρ(ωg ) + −1∂ ∂f
for some smooth real valued function f . In particular, every expanding Kähler Ricci soliton on a compact
Kähler manifold is Einstein since compact Kähler manifolds with negative first Chern class do not admit
non-trivial holomorphic vector fields.
For more background on Kähler Ricci solitons see [CH00], [Cao97], [Cao96], [Bry08], and [FIK03].
5.3. STATEMENT OF THE EXISTENCE THEOREM
43
5.3. Statement of the Existence Theorem
The goal of this chapter is summarized in the following theorem:
Theorem 5.3.1 (Main Theorem). Let (C, JC , gC ) be a Ricci flat Kähler cone. Suppose that π : M → C
is a simply connected equivariant resolution1 of C
√ with exceptional set E and X the lift of the radial
holomorphic vector field, i.e. X|M \E = 21 π ∗ (r∂r − −1Jr∂r ). Then there exists an expanding (gradient)
Kähler Ricci soliton metric on M with the soliton vector field X if there exists a hermitian metric h on
KM such that the associated curvature form Θh is positive and L=(X) Θh = 0.
Remark 5.3.2. Roughly speaking, what the theorem says is that any equivariant resolution π : M → C
of a Ricci flat Kähler cone C such that c1 (M ) < 0 admits an expanding Kähler Ricci soliton metric.
This should be compared to the results of [Got12] that any crepant resolution (c1 (M ) = 0) of a Ricci
flat Kähler cone admitds a unique Ricci flat Kähler metric in every Kähler class.
Remark 5.3.3. By theorem 4.14 in [vC12] the link of such a cone C which is a Sasaki Einstein manifold
cannot be simply connected unless it is the round sphere S 2n−1 with the standard CR-structure, i.e.
C = Cn with the euclidean metric2.
Here we will prove the following theorem which implies the main theorem stated above but has the
advantage of using maybe more familiar terminology from Riemannian geometry
Theorem 5.3.4 (Existence of expanding Kähler Ricci Solitons, 1st Version). Let (M, J, g) be a Kähler
manifold asymptotic to a Ricci flat Kähler cone (C, JC , ωC ) at rate r−τ , τ > 2. Moreover let X be a
holomorphic vector field on M such that
√
∂
∂
− −1Jr ∂r
) outside a compact set.
(1) X = 21 (r ∂r
(2) <(X) = ∇θX for some real valued function θX .
(3) L=(X) ω = 0.
If there exists a function F ∈ C ∞ (M ) with L=(X) F = 0 such that
√
¯
(5.4)
ρ(ωg ) + ωg = LX ωg + −1∂ ∂F,
√
0
¯ for some ϕ ∈ U such that
then there exists a complete Kähler metric g with ωg0 = ωg + −1∂ ∂ϕ
ωg0 + ρ(ωg0 ) = LX ωg0 .
Proof of the main theorem 5.3.1 from theorem 5.3.4. Suppose there exists a hermitian met2
ric h on KM as in theorem 5.3.1 and set ω0 := 2πΘh . Since [Θh ] ∈ H√
c (M ) using proposition B.2.3
¯ L=(X) ωg = 0, and
we can find a complete Kähler metric g on M such that ωg = ω0 + −1∂ ∂ϕ,
∗
ωg = π ωC outside a compact
set. From chapter 2 we know that there exists a smooth function f such
√
¯ . And we can conclude that
that ρ(ωg ) + 2πΘh = −1∂ ∂f
√
√
√
¯ − 2πΘh + −1∂ ∂f
¯ = −1∂ ∂(f
¯ + ϕ).
ωg + ρ(ωg ) = ω0 + −1∂ ∂ϕ
Let F := f + ϕ − θX , where θX is the potential of X with respect to g then
√
¯
ωg + ρ(ωg ) = LX ωg + −1∂ ∂F
and the existence of the soliton Kähler metric follows from theorem 5.3.4.
Remark 5.3.5. The two theorems are in fact equivalent. Suppose there exists a Kähler metric ωg and a
function F as in theorem 5.3.4. From the discussion in [vC12] it follows that M is in fact a resolution of
C ∪ {o} (see remarks below). Now ωg induces a hermitian metric h0 on KM such that ρ(ωg ) = −2πΘh0 .
Hence for h = exp(θX + F )h0
√
¯ = −ωg ,
−2πΘh = ρ(ωg ) − LX ωg − −1∂ ∂F
meaning that h satisfies all the assumptions of theorem 5.3.1.
1recall that by equivariant resolution we mean that radial vector field has a holomorphic lift
2More precisely, the cone cannot be Gorenstein (which is implied by simply connectedness of S but in fact a weaker
condition), unless it is Cn .
44
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
5.3.1. Reduction to a Monge Ampere equation. Assume that we can find a Kähler
√ Ricci
¯
soliton metric ω 0 := ωg0 such that L=(X) ω 0 = 0 and ω := ωg as in theorem 5.3.4. Then ω 0 = ω + −1∂ ∂ϕ
i.e. ω 0 is in the same Chern Bott class as ω (see below). We can subtract the equations for ω 0 and ω to
obtain
√
¯
ω 0 − ω + ρ(ω 0 ) − ρ(ω) = LX ω 0 − LX ω − −1∂ ∂F.
And so
0n √
√
√
√
ω
¯
¯
¯ − −1∂ ∂F.
¯
= −1LX ∂ ∂ϕ
−1∂ ∂ϕ − −1∂ ∂ log
n
ω
√
√
¯ = −1∂ ∂X(ϕ)
¯
By lemma 5.4.23 below we have −1LX ∂ ∂ϕ
and the equation becomes
0n √
√
√
√
¯ − −1∂ ∂¯ log ω
¯
¯
−1∂ ∂ϕ
= −1∂ ∂X(ϕ)
− −1∂ ∂F.
ωn
Up to a pluriharmonic function ϕ satisfies a scalar partial differential equation
0n ω
−ϕ + log
+ X(ϕ) = F.
ωn
Conversely if the equation
ωϕn
−ϕ + log
+ X(ϕ) = F.
ωn
√
¯ is a soliton metric if ω = ωg solves (5.4).
has a solution then clearly ω 0 = ω + −1∂ ∂ϕ
5.3.2. Reduction of the assumptions of theorem 5.3.4. To prove the theorem we show first
that we can make the following assumptions without loss of generality:
(1) The function F can be assumed to have compact support
(2) Φ∗ ω = ωC outside of a compact set
(3) θX = r2 outside of a compact set.
We first show that the assumptions of the theorem imply that we can choose a background Kähler
metric as in the theorem that is equal to the cone metric outside of a compact set (assumption (2)) and
that F has compact support (assumption (1)). This can be seen as follows. Note first that Φ can be
extended across K to become a degree one morphism Φ̃ : M → C ∪ {o} making (M, C, Φ̃) a resolution
of C ∪ {o} (see Appendix B). We have the following expression for the canonical divisor
X
KM = Φ̃∗ KC +
aj Ej , aj ∈ R
j
p
where KC is in fact Q-Cartier in C(S) ∪ {o} (ı∗ OC(S) (KC
) is an invertible sheaf for some p ∈ N) and,
for some p ∈ N, pKC is trivial. It follows that KM is generated by the prime elements Ei of the
exceptional divisor E which implies that c1 (M ) ∈ Hc2 (M ). Since we assume that [ω] = −2πc1 (M ) we
have [ω] ∈ Hc2 (M ) and so by proposition B.2.3 there exists a Kähler form ω̃ with L=(X) ω̃ = 0 such that
Φ∗ ω̃ = ωC outside of a compact set. Here is another way to see that the existence of ω ∈ Hc2 (M ) is
necessary. As g 0 is a solution of the soliton equation and the curvature decays quadratically it is by
theorem 4.3.1 also a solution of the Ricci flow starting from the cone C. But since the initial cone C is
Ricci flat the solution, g 0 , by theorem 3.3.1 approaches the cone metric in fact exponentially fast. By
proposition 2.8.3 this means that [ω 0 ] ∈ Hc2 (M ) which again by proposition B.2.3 implies the existence
of ω̃ with [ω̃] = [ω 0 ]. If the initial cone is not Ricci flat we cannot assume that c1 (M ) ∈ Hc2 (M ) in
general.
To show that we can assume (1) note first that (2) implies that F is pluriharmonic outside some
compact set. In particular F is pluriharmonic on C \ BR for some large R > 0. This implies that X(F )
is holomorphic. But since X(F ) is real valued X(F ) = c0 for some constant c0 outside of BR . Hence,
on C \ BR , F = c0 log(r) + c1 where c1 is a function that depends only on x ∈ S. But we also have
∆gC F = 0 which implies that (2n − 2)r−2 c0 + r−2 ∆gS c1 = 0. Integrating over S shows that c0 = 0 and
hence F = c1 = C for some constant C. Clearly F − C can be extended across the compact set. Finally
we can replace F by F − C to get a smooth function on M with compact support that satisfies (5.4).
Assumption (3) can be shown similarly.
5.3. STATEMENT OF THE EXISTENCE THEOREM
45
5.3.3. Weighted spaces. From now on we will assume that (M, J, g) is as in theorem 5.3.4 and
in addition g = Φ∗ gC outside of a compact set and F is compactly supported. To show that the
Monge Ampere equation derived above has a solution we will employ the continuity method. Part of
this method involves showing that the linearization of a non-linear operator is an isomorphism between
certain function spaces X and Y. This motivates definition of the following spaces:
(1) The weighted Hölder space on M
k,α
k,α
Cw
(M ) := {u ∈ Cloc
(M ) : kukCwk,α (M ) < ∞},
kukCwk,α (M ) (M ) := k exp(θX )ukC k,α (M ) .
k,α
Note that (Cw
(M ), k · kC k,α (M ) ) = (e−θX C k,α (M ), k · kC k,α (M ) ).
(2) The space of decaying (k, α)-regular Kähler potentials
√
k,α
¯ > 0}
Kk,α := {u ∈ Cw
(M ) : ω + −1∂ ∂u
and the space of smooth decaying Kähler potentials K := ∩k Kk,α (M ).
(3) The space of Jζ = =(X)-invariant C k,α -functions
k,α
CX
(M ) := {u ∈ C k,α (M ) : LJζ u = 0}.
(4) The space of Jζ-invariant decaying functions
k,α
k,α
k,α
Cw,X
(M ) := CX
(M ) ∩ Cw
(M ).
(5) The space (X k,α , k · kX k,α )
k,α
(M ) : kX(eθX u)kC k−2,α < ∞}
X k,α := {u ∈ Cw,X
together with the norm
kukX k,α := kukCwk,α (M ) + kX(eθX u)kC k−2,α (M ) .
(6) The space (X̃ k,α , k · kX̃ k,α )
k,α
X̃ k,α := {u ∈ CX
(M ) : kX(u)kC k−2,α < ∞} = eθX X k,α
together with the norm
kukX k,α := kukC k,α (M ) + kX(u)kC k−2,α (M ) .
(7) The space of Jζ-invariant decaying Kähler potentials in X k,α :
U k,α := X k,α ∩ Kk,α
and
U := ∩k U k,α .
Remark 5.3.6. The reason that we require X(eθX u) ∈ C k−2,α will become clear later. We will consider
k,α
second order operators on Cw
(M ) (whose linearization is) of the form L = ∆+X (∆ being the Laplacian
with respect to g). The idea is to conjugate L with eθX to obtain an operator L̃ := eθX ◦ L ◦ e−θX which
acts on the unweighted space C k,α . In particular we will have a term of the form
eθX X(e−θX ũ) = −|X|2 ũ + X(ũ).
On the other hand
eθX ∆(e−θX ũ) = ∆ũ + (|X|2 − ∆θX )ũ − 2X(ũ).
So
L̃(ũ) = ∆ũ − X(ũ) − (∆θX )ũ.
k−2,α
Now it is clear that the image of u under L lies in Cw
if and only if the image of ũ := eθX u
k−2,α
under L̃ lies in C
. But this is quite obvious for the first and third term above which means that
X(ũ) = X(eθX u) has to be an element of C k−2,α .
46
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
5.3.4. Restatement of the main theorem. Using the reduction to a Monge Ampere equation
above we can now state the existence theorem as scalar partial differential equation:
Theorem 5.3.7 (Existence of expanding Kähler Ricci Solitons, PDE Version). Let (M, J, g) be a Kähler
manifold which is biholomorphically isometric to a Ricci flat cone (C, JC , ωC ) outside a compact set.
Moreover let X be a holomorphic vector field on M such that
√
∂
∂
(1) X = 21 r ∂r
− −1Jr ∂r
outside a compact set.
2
(2) <(X) = ∇θX , θX = r outside a compact set.
(3) L=(X) ω = 0.
Then for any F ∈ Cc∞ (M ) there exists a unique ϕ ∈ U such that ϕ satisfies the equation
ωϕn = eϕ−X(ϕ)+F ω n .
(5.5)
Remark 5.3.8. The assumptions in the theorem actually imply that M is an equivariant resolution of
C ∪ {o}.
Remark 5.3.9. Note that the Kähler Ricci flow associated to our expanding Ricci soliton converges
smoothly on complements of open neighborhoods of the mentioned compact set to the original Ricci flat
Kähler cone. Moreover, it converges in the Gromov Hausdorff sense as t → 0 to the metric completion of
(C, gC ) as a metric space. This flow is different from the one constructed in [ST09] (see also [BCG+ 12]).
There Ricci flows on compact varieties with log terminal singularities are constructed. In particular, following the argument in [ST09] and [BCG+ 12] the corresponding Kähler Ricci flow would be constructed
as follows: suppose C ∪ {o} is a log terminal singularity and π : M → C ∪ {o} a minimal
√ resolution
¯ · , t),
of C. Then there exists a unique Ricci flow ωϕ on M such that ωϕ = π ∗ θ0 + tπ ∗ χ + −1∂ ∂ϕ(
p ∗
where in our case θ0 = 0 and 2π π χ is a curvature form corresponding to a hermitian metric h on the
p
well-defined line bundle π ∗ KC
(p ∈ N sufficiently large) on M which in our case is trivial and hence can
be chosen to be equal to zero. Finally ϕ( · , t) solves
n
ωϕ
∂ϕ
= log
, (x, t) ∈ C × (0, T ]
∂t
µh
where log(µh ) is a potential for χ (in the terminology of [BCG+ 12] µh is called the associated adapted
measure) and ϕt → ϕ0 = r2 as t → 0 uniformly in complements of open neighborhoods of {o}. It follows
that the unique solution is the stationary solution on the cone. Of course the cone is not compact as
required in [ST09]. However, it is not hard to give compact examples (compactifications of the examples
constructed in [FIK03]) with log-terminal singularities modeled on Ricci flat Kähler cones which have
solutions of type described in [ST09] as well as solutions on topologically different manifolds.
Note that in the program of Song and Tian the singularities that form during the Kähler Ricci flow
are not necessarily log-terminal. In particular the limit might not be a variety with Q-Gorenstein
singularities (for example for a small, i.e. higher codimension contraction) and hence their tangent cones
are certainly not Ricci flat. So to study such initial conditions one would have to investigate general
Kähler cones where the canonical line bundle is not well-defined as above. In this case it is conjectured
that the Ricci flow can be continued only after a flip and the blow-up limit of the resulting flow should
be an expanding Kähler Ricci soliton coming out of a cone. While these expanding Kähler Ricci solitons
are also topologically different from the initial cone their exceptional locus has higher codimension and
so the curvature can decay at most quadratically. In particular, these expanding Kähler Ricci solitons
do not fall into the class of expanding Kähler Ricci solitons constructed here. In conclusion it seems
likely (at least on complex surfaces) that the static solution of the Ricci flow on the Ricci flat cone is
indeed the preferred solution while the solutions presented here seem to correspond to Toppings reverse
bubbling examples for the harmonic map heat flow (i.e. the entropy jumps down at time 03).
5.4. Proof of the existence theorem
We begin by stating four theorems which we then combine to prove theorem 5.3.7.
3Note however that λ (g
nc FIK ) > 0 while λnc (gCao ) = λnc (geucl ) = 0
5.4. PROOF OF THE EXISTENCE THEOREM
47
Theorem 5.4.1. [a priori estimates I] Let (M, J, g) and X as in theorem (5.3.7). For any D1 > 0 there
∞
exists D2 , D3 , D4 , D5 such that if F ∈ Cc∞ (M ) ∩ CX
(M ), kF kCw1,α ≤ D1 , diam(supp(F )) ≤ D1 , and
3,α
ϕ∈U
a solution of
ωϕn = eϕ−X(ϕ)+F ω n
¯ C 0 (M ) ≤ D4 , and k∇∂ ∂ϕk
¯ C 0 (M ) ≤ D5 .
then ϕ ∈ C ∞ (M ), kϕkC 0 (M ) ≤ D2 , kX(ϕ)kC 0 (M ) ≤ D3 , k∂ ∂ϕk
loc
w
Theorem 5.4.2. [a priori estimates II] Let (M, J, g) and X as in theorem (5.3.7). For any D1 , D2 , D3 , D4 , D5 >
∞
0 there exists D6 , D7 such that if F ∈ Cc∞ (M ) ∩ CX
(M ), kF kCw1,α ≤ D1 , diam(supp(F )) ≤ D1 and
ϕ ∈ U 3,α a solution of
ωϕn = eϕ−X(ϕ)+F ω n
¯ C 0 (M ) ≤ D4 , k∇∂ ∂ϕk
¯ C 0 (M ) ≤ D5 then kϕkX 3,α (M ) ≤
with kϕkCw0 (M ) ≤ D2 , kX(ϕ)kC 0 (M ) ≤ D3 , k∂ ∂ϕk
D6 .
Theorem 5.4.3. Let (M, J, g) and X be as in theorem (5.3.7). Suppose that ϕ ∈ U 3,α is a solution of
ωϕn = eϕ−X(ϕ)+F ω n
then ϕ ∈ U k,α for all k ∈ N, α ∈ (0, 1).
Theorem 5.4.4. Let (M, J, g) and X be as in theorem (5.3.7). Suppose that F̃ ∈ Cc∞ (M ). And ϕ̃ ∈ U 3,α
is a solution of
ωϕ̃n = eϕ̃−X(ϕ̃)+F̃ ω n .
(5.6)
Then for any F ∈ Cc∞ (M ) such that kF − F̃ kCw1,α (M ) is sufficiently small there exists a solution ϕ ∈ U
of
ωϕn = eϕ−X(ϕ)+F ω n
Theorem 5.4.5. Let (M, J, g) and X be as in theorem 5.3.7 and F ∈ Cc∞ (M ). Then there exists at
most one solution ϕ ∈ U 3,α of
ωϕn = eϕ−X(ϕ)+F ω n .
The partial differential equation (5.5) we consider is a very familiar one. In fact, it is almost the same as
the Monge Ampere equation associated with the Calabi conjecture (up to the linear term X(ϕ) − ϕ), the
problem of finding a Kähler Einstein metric with negative scalar curvature (up to the term X(ϕ)), or
shrinking Kähler Ricci solitons (up to the sign in front of ϕ). If M is compact the existence of solutions
in the first two cases has been shown by Yau and Aubin. In principle our partial differential equation
(5.5) is most similar to the Kähler Einstein problem (although the asymptotic behavior of the solution
is very different). The essential part of the proof is to find uniform a priori estimates for ϕ and just
like in the Kähler Einstein problem we can easily find uniform C 0 -estimates for ϕ using the maximum
principle. However, here we have to deal with two additional problems: the non-compactness of M and
the fact that the vector field X is unbounded. The first problem can be resolved by using the weighted
function spaces introduced earlier. Here these weighted spaces are much simpler than those considered
for example in the partial differential equation associated to the non-compact Calabi Yau problem. The
second obstacle (unbounded coefficients) requires a bit more work and basically uses results for linear
elliptic operators with unbounded coefficients and global Schauder estimates developed there.
So here are the basic ideas of the proof: as usual we show the existence of a solution using the continuity
method. Openness (theorem 5.4.4) will follow from the fact that the linearized operator4 ∆ − X − λ
acting on weighted spaces is invertible for λ > 0. In the compact case this is easy since the choice of
function spaces is quite obvious. In the non-compact case the function spaces have to chosen a bit more
carefully. If the underlying cone is Ricci flat then it turns out that there exists a simple choice (the
weighted spaces from above) and the problem becomes very similar to the compact case. The closedness
will be shown by establishing the following a priori estimates (theorem 5.4.1): first the unweighted
C 0 -estimates follow directly from the maximum principle. The weighted C 0 -estimates are also obtained
4Note that there is a minus sign in front of X. In fact, this operator is not the linearization of the Monge Ampere operator
associated with (5.5) but its conjugation with exp(−θX ) as described in the previous subsection.
48
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
using the maximum principle by considering the function exp(θX )ϕ on the conical end. In combination
0
with the C 0 -estimates in the interior set this gives the desired Cw
-estimates. Before we can proceed
2
to the so-called C -estimates we have to establish uniform estimates for X(ϕ).
√ These follow from the
¯ > 0. Next we find
weighted C 0 -estimates and the the fact that ϕ is a Kähler potential, i.e. ω + −1∂ ∂ϕ
¯
uniform estimates for |∂ ∂ϕ|.
This is done exactly like in Aubin’s and Yau’s work (up to some additional
terms), i.e. again using the maximum principle. Note that this does not yet give a C 2 -estimate for ϕ
directly. However, together with the estimates for X(ϕ) and ϕ it follows that the metric g 0 associated
¯ are again due to Yau. This implies in
to ωϕ is uniformly equivalent to g. The estimates for ∇∂ ∂ϕ
particular uniform estimates for kgϕ−1 kC 0,α . Next the maximum principle establishes decay estimates
0
¯ (at the same
for X(ϕ) which are slightly weaker than Cw
. This, in turn, gives us decay estimates for ∂ ∂ϕ
2,α
rate). And finally the global Schauder estimates for the conical end provide us with the Cw
-estimates
k,α
and a bootstrapping argument delivers the Cw
-estimates for k > 2 (theorems 5.4.2 and 5.4.3).
Proof of theorem 5.3.7. For t ∈ [0, 1] denote by ϕt ∈ U 3,α a solution (if one exists) of
ωϕn = eϕ−X(ϕ)+tF ω n .
And let
S := {s ∈ [0, 1] : there exists a solution ϕt ∈ U 3,α for all t ∈ [0, s]}.
Theorem 5.4.6. The set S is open in [0, 1].
Proof. Let t0 ∈ S and choose > 0 small enough such that kF kCw1,α (M ) is sufficiently small to
apply theorem 5.4.4. Then by theorem 5.4.4 we get a solution ϕt ∈ U 3,α for any t0 ≤ t < t0 + and so
it follows that s ∈ S for any s < t0 + , in other words S is open.
Theorem 5.4.7. The set S is closed in [0, 1].
Proof. We have to show that for any sequence tj ∈ S which converges in [0, 1] the limit limj→∞ tj =
t is also contained in S. So assume that ϕj = ϕtj is a sequence of solutions of
ωϕnj = eϕj −X(ϕj )+Fj ω n ,
∞
where Fj = tj F . Clearly Fj ∈ Cc∞ (M ) ∩ CX
(M ) and there is a positive number D1 such that
kFj kCw1,α (M ) ≤ D1 and diam(supp(Fj )) ≤ D1 for all j. By theorems 5.4.1 and 5.4.2 we get a uniform bound D6 for kϕj kX 3,α (M ) . By Arzela Ascoli we can find a subsequence ϕjk that converges in
3,β
3,α
(M ), and since the sequence is bounded in X 3,α we have ϕ ∈ X 3,α .
Cloc
(M ) to a function ϕ ∈ Cw
3,α
Moreover, ϕ ∈ U . Indeed, assume to the contrary that ϕ ∈
/ U 3,α . Then there exists a point x ∈ M
n
such that ωϕ (x) = 0 (since ωϕ > 0 outside a large compact set by the decay property of ϕ). But this
means that limj→∞ ϕj (x) − X(ϕj )(x) + tj F = −∞ which contradicts the uniform boundedness of ϕj
and X(ϕj ). Therefore ϕ satisfies
ωϕn = eϕ−X(ϕ)+tF ω n .
And so t ∈ S which proves that S is closed.
Now we can finish the proof of theorem 5.3.7. Since S is non-empty as 0 ∈ S (with ϕ0 = 0) and S is
closed and open in [0, 1] we have S = [0, 1]. This means that there exists a solution ϕ1 ∈ U 3,α which is
exactly the desired solution ϕ of (5.5).
5.4.1. Proof of theorems 5.4.1-5.4.5.
Proof of theorem 5.4.1. We will now prove theorem 5.4.1. Parts of this proof are very similar
¯ and ∇∂ ∂ϕ
¯ are almost exactly
to the compact case. In particular the proofs of the estimates for ϕ, ∂ ∂ϕ
0
the same as in [Yau78]. However, we will need in addition uniform estimates for ϕ in Cw
(M ) and X(ϕ)
in order to carry out the second and third order estimates of Yau and Aubin.
So in the following five subsections we assume that ϕ ∈ U 3,α is a solution of
ωϕn = eϕ−X(ϕ)+F ω n ,
with F ∈ Cc∞ (M ) and kF kCw1,α (M ) ≤ D1 and diam(supp(F )) ≤ D1 .
5.4. PROOF OF THE EXISTENCE THEOREM
49
5.4.1.1. Local regularity. In order to start with the proof of theorem 5.4.1 we will need the following
proposition
Proposition 5.4.8. Let (M, J, g) be as in theorem 5.4.1, k ≥ 1, and ϕ ∈ U 3,α a solution of (5.5) with
k,α
k+2,α
F ∈ Cloc
(M ). Then ϕ ∈ Cloc
(M ).
√
¯
¯ 0 , u0 ∈ C ∞ (U ),
Proof. By the local ∂ ∂-lemma
[Joy00] ω can be locally expressed as −1∂ ∂u
loc
U ⊂ M open, we can set u := u0 + ϕ and study the equation in local holomorphic coordinates on a
possibly smaller subset Ω ⊂ U
log(det(ui̄ )) + X(u) − u = F̃
k,α
for F̃ ∈ Cloc
(Ω), Ω ⊂ Cn . Now u ∈ C 3,α (Ω) by assumption and uk := ∂zk u satisfies a linear elliptic
partial differential equation with coefficients in C 1,α (Ω). Indeed
∂ 2 uk
∂F
= gϕi̄
.
∂zk
∂zi ∂ z̄̄
And since F̃ ∈ C 2,α (Ω) it follows form standard regularity theory that ∇u ∈ C 3,α (Ω0 ) for Ω0 ⊂ Ω and
4,α
hence ϕ ∈ Cloc
(M ). The case k > 1 works similarly.
5.4.1.2. C 0 -estimates. We start with the uniform estimates for ϕ. These are very easy to obtain as
the sign in front of the zeroth order term in the equation helps us to apply the maximum principle in a
straightforward fashion.
Lemma 5.4.9. Let ϕ ∈ U 3,α be a solution of (5.5). Then there exists a constant C1 = C1 (g) such that
kϕkC 0 ≤ C1 .
Proof. By assumption limx→∞ ϕ(x) = 0. So if sup ϕ > 0 (inf ϕ < 0) then there must be a point
xmax ∈ M (xmin ∈ M ) such that ϕ(xmax ) = sup ϕ (ϕ(xmin ) = inf ϕ). But at xmax (xmin ) we have
ωϕn ≤ ω n (ωϕn ≥ ω n ) and X(ϕ)(xmax ) = 0 (X(ϕ)(xmin ) = 0). And so
sup ϕ ≤
−F (xmax ) ≤ − inf F ≤ sup |F |
inf ϕ ≥
−F (xmin ) ≤ − sup F ≤ − sup |F |,
and hence
kϕkC 0 ≤ C1 .
0
5.4.1.3. Cw
-estimates. Unlike in the compact case and in the non-compact ALE case of the CalabiYau theorem described in [Joy00] we have to find weighted C 0 estimates for ϕ first in order to proceed
to the higher order estimates. In particular, we will need the decay estimates to get uniform bounds for
X(ϕ).
Lemma 5.4.10. Let ϕ ∈ U be a solution of (5.5). Then there exists a uniform constant C2 = C2 (D1 , g)
such that
kϕkCw0 ≤ C2 .
Proof. We split the proof into two parts. First we prove uniform upper bounds for exp(θX )ϕ. This
uses the fact that exp(µθX )ϕ is actually a subsolution of a linear elliptic partial differential equation
for any µ ∈ (0, 1). Again the maximum principle gives us uniform estimates for sup exp(µθX )ϕ for
any µ ∈ (0, 1). As these estimates do not depend on µ we obtain the desired upper bound. To prove
that inf exp(αθX )ϕ is bounded from below for any α ∈ (0, 1) we first construct a barrier function χ
(independent of ϕ) on the cone end which decays at the rate exp(−µθX ) for some small µ > 0 that
bounds ϕ from below outside a uniformly large compact set. Here we will need the C 0 estimates for
ϕ from before. In a second step we apply the minimum principle to the function exp(αθX )ϕ for any
α ∈ (0, 1) which again gives us a uniform lower bound independent of α. Arguing as above it follows
50
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
that exp(θX )ϕ is uniformly bounded form below and hence kϕkCw0 (M ) ≤ C2 for some uniform constant
C2 . Note first that
n
Z 1Z σ
ωϕ
¯ 2 dτ dσ + X(ϕ) − ϕ
|∂ ∂ϕ|
F = log
+ X(ϕ) − ϕ = ∆ϕ −
ωτ ϕ
ωn
0
0
≤ ∆ϕ + X(ϕ) − ϕ.
In other words ϕ is a subsolution of a linear elliptic equation. This uses the concavity of the Q 7→
log(det(1 + Q)). Let µ ∈ (0, 1) and ψµ := eµθX ϕ. Then ψµ satisfies
F eµθX ≤ ∆ψµ + (1 − 2µ)X(ψµ ) − (µ∆θX + (µ − µ2 )|X|2 + 1)ψµ .
Since F has compact support we can find a uniformly large compact set K such that on M \ K (using
∆θX = n on M \ K)
0 ≤ ∆ψµ − X(ψµ ) − (µn + (µ − µ2 )|X|2 + 1)ψµ
and since µn + (µ − µ2 )|X|2 + 1 ≥ 1 and ψµ (x) → 0 as x → ∞ (here we use that µ < 1) it follows by
the maximum principle that
sup ψµ ≤ C20 (K).
M
And since
C20 (K)
does not depend on µ < 1 we have, letting µ → 1,
sup eθX ϕ ≤ C20 (K).
M
To show the lower bound we have to work a bit harder. First recall that outside of a compact set K0
(i.e. where ω is exactly the cone metric and θX = r2 ) we have
√
¯ X and |X|2 = θX .
ω = −1∂ ∂θ
So let us consider the set
Ωκ := {x ∈ M : θX > κ}.
We know that there exists a uniformly large κ0 such that Ωκ ⊂ M \ K0 and F = 0 on Ωκ for κ ≥ κ0 .
On such a set Ωκ we will to construct a function χ such that
n
ωχ
log
+ X(χ) − χ ≥ 0
ωn
on Ωκ and χ ≤ ϕ on ∂Ωκ . We use the following ansatz
χ := χρ,µ = −ρe−µθX
where ρ > 0, µ ∈ (0, 1). We cannot expect that χ ∈ U (i.e. ωχ > 0) on all of M . However, we show
that for an appropriate choice of κ we will have ωχ > 0 on Ωκ . This works for example for the following
choices of ρ, µ, and κ (C1 being the constant from the the previous lemma):
κ := max{4C1 e−1/2 , κ0 },
µ :=
3
,
2κ
ρ := C1 exp(3/2).
First note that on ∂Ωκ we have θX = κ and so on ∂Ωκ
χ = −C1 exp(3/2) exp(−3/2) = −C1 ≤ ϕ.
The eigenvalues of the hermitian 2-tensor associated to
√
√
¯ = (1 + ρµe−µθX )ω − −1ρµ2 e−µθX ∂θX ∧ ∂θ
¯ X
ωχ = ω + −1∂ ∂χ
are (1+ρµ exp(−µθX )) with multiplicity (n−1) (the base eigenvalue i.e. the ones whose eigenvectors are
transversal to X) and (1 + ρ(µ2 |X|2 − µ) exp(−µθX )). Clearly the transversal eigenvalues are bounded
5.4. PROOF OF THE EXISTENCE THEOREM
51
from below by 1. As for the remaining eigenvalues we have in Ωκ
(1 − ρ(µ2 |X|2 − µ) exp(−µθX )) = 1 − C1 exp(3/2)µ(µ|X|2 − 1) exp(−µθX )
≥ 1 − C1 exp(3/2)µ sup (y − 1) exp(−y)
y>3/2
= 1 − C1 exp(3/2)
= 1 − C1
≥1−
>
3
exp(−2)
2κ
3 −1/2
e
2κ
3
8
1
,
2
So on Ωκ we have
ωχn
≥ log(1 − ρ(µ2 |X|2 − µ) exp(−µθX )).
log
ωn
By the estimates above we know that on Ωκ
ρ(µ2 |X|2 − µ) exp(−µθX ) ∈ (0, 1/2).
Now we can use that
log(1 − x) ≥ −2x,
if x ∈ (0, 1/2)
to conclude that in Ωκ .
log(1 − ρ(µ2 |X|2 − µ) exp(−µθX )) ≥ −2ρ(µ2 |X|2 − µ) exp(−µθX ).
We also have in Ωκ
X(χ) − χ = ρ(µ|X|2 + 1)e−µθX .
Putting everything together
n
ωχ
log
+ X(χ) − χ ≥ −ρ((2µ2 − µ)|X|2 − 2µ − 1)e−µθX
ωn
So on Ωκ (if µ < 1/2, which we can assume without loss of generality by choosing C1 large if necessary)
n
ωχ
log
+ X(χ) − χ ≥ 0.
ωn
and
log
ωϕn
ωn
+ X(ϕ) − ϕ = 0.
Letting ψ := χ − ϕ we have on Ωκ
√
¯ n
(ωϕ + −1∂ ∂ψ)
log
+ X(ψ) − ψ ≥ 0.
ωϕn
And so by the maximum principle
sup ψ ≤ max{0, sup(χ − ϕ)} ≤ 0.
Ωκ
∂Ωκ
It follows that χ ≤ ϕ in Ωκ and by the observations above we have a lower bound for ϕ of the form
−ρ exp(−µθX ).
It remains to show that we can find a uniform constant C200 such that ϕ ≥ −C200 exp(−θX ). Again let
ψα := exp(αθX )ϕ, α ∈ (0, 1). We can choose a uniformly large compact set K outside of which
√
√
¯ X + −1α2 ϕ∂θX ∧ ∂θ
¯ X > 0.
ω − −1αϕ∂ ∂θ
So if ψα has a minimum in M \K then at this point (call it pα and ϕ(pα ) = mα and ψ(pα ) = inf ψα = m0α )
√
√
√
¯
¯ X (pα ) + −1α2 mα ∂θX (pα ) ∧ ∂θ
¯ X (pα ) > 0
−1αmα ∂ ∂θ
ω(pα ) + −1∂ ∂ϕ(p
α) ≥ ω −
52
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
and so
log
(ω −
√
√
¯ X + −1α2 mα ∂θX ∧ ∂θ
¯ X )n −1αmα ∂ ∂θ
− α|X|2 mα − mα ≤ 0.
ωn
Again it follows that (calculating the eigenvalues)
0 ≥ (n − 1) log(1 − αmα ) + log(1 − αmα + α2 |X|2 mα ) − α|X|2 mα − mα
≥ log(1 − αmα + α2 |X|2 mα ) − α|X|2 mα − mα
where we assume that mα ≤ 0. Moreover without loss of generality we can assume that
(α2 |X|2 − α)mα ≥ −
1
2
on M \ K and since
log(1 − x) ≥ −x − x2 ,
x ∈ (0, 1/2)
we have (again for a sufficiently large choice of K independent of α)
0 ≥ (α2 |X|2 − α)mα − (α2 |X|2 − α)2 m2α − α|X|2 mα − mα
≥ (α2 |X|2 − α)mα + (α2 |X|2 − α)2 C exp(−µθX ) mα − α|X|2 mα − mα .
{z
}
|
<1/2
This implies
1
0 ≥ (α2 |X|2 − α)m0α − m0α − α|X|2 m0α
2
hence ψα ≥ 0 on M \ K and we can find a constant C200 = C200 (K) such that
ϕ ≥ −C200 exp(−αθX ).
Since C200 and K do not depend on α we also have
ϕ ≥ −C200 exp(−θX ).
5.4.1.4. C 0 -estimates for |X(ϕ)|. Now we estimate |X(ϕ)|. Unlike in the compact case this estimate
has to be done separately since X does not have bounded coefficients. We start with some basic
observations. Recall that =(X)(ϕ) = 0 and we denote ζ := <(X).
Lemma 5.4.11. Let ϕ be a smooth function such that Jζ(ϕ) = 0. Then Jζ(ζ(ϕ)) = 0.
Proof. Note that
Jζ(ζ(ϕ)) = [Jζ, ζ](ϕ) + ζ(Jζ(ϕ)) = −Lζ (Jζ)(ϕ) = −Lζ Jζ + JLζ ζ(ϕ) = 0,
where we used that Lζ J = 0 since ζ is a real holomorphic vector field.
Lemma 5.4.12. Let ϕ ∈ U 3,α . Then for any vector field Y on M we have
√
¯
−1∂ ∂ϕ(Y,
JY ) > −|Y |2 .
Proof. By assumption ωϕ (Y, JY ) > 0 and so
√
√
¯
¯
0 < ω(Y, JY ) + −1∂ ∂ϕ(Y,
JY ) = |Y |2 + −1∂ ∂ϕ(Y,
JY )
which proves the lemma.
Lemma 5.4.13. Suppose ϕ ∈ U 3,α . Then
X(X(ϕ)) =
√
¯
−1∂ ∂ϕ(X,
X̄)
and in particular
X(X(ϕ)) ≥ −|X|2 .
5.4. PROOF OF THE EXISTENCE THEOREM
53
Proof. Note first that by the previous lemma
X(X(ϕ)) = ζ(ζ(ϕ)) = X̄(X(ϕ))
where X̄ is the complex conjugate of X. Now as usual we denote
√
√
√
√
∂
∂
∂
1
∂
∂
1
∂
dzi = dxi + −1dyi , dz̄̄ = dxj − −1dyj ,
=
− −1
,
=
+ −1
,
∂zi
2 ∂xi
∂yi
∂ z̄̄
2 ∂xj
∂yj
moreover
ζ=
n
X
i=1
and so
Xi = ζi +
√
ζi
∂
∂
+ ζ n+i
∂xi
∂yi
−1ζ n+i , X̄ ̄ = ζ j −
√
−1ζ n+j .
To make notation easier we let
Aij :=
Then
√
∂2ϕ
∂2ϕ
∂2ϕ
∂2ϕ
+
, Bij :=
−
.
∂xi ∂xj
∂yi ∂yj
∂xi ∂yj
∂yi ∂xj
¯
−1∂ ∂ϕ(ζ,
Jζ) = Aij ζ i ζ j + ζ n+i ζ n+j + Bij ζ i ζ n+j − ζ n+i ζ j .
On the other hand
X i X̄ ̄
√
√
√
∂2ϕ
= (ζ i + −1ζ n+i )(ζ j − −1ζ n+j )(Aij + −1Bij )
∂zi ∂ z̄̄
= Aij ζ i ζ j + ζ n+i ζ n+j + Bij ζ i ζ n+j − ζ n+i ζ j .
But since
X̄(X(ϕ)) = X i X̄ ̄
it follows that X(X(ϕ)) =
√
∂2ϕ
∂zi ∂ z̄̄
¯
−1∂ ∂ϕ(ζ,
Jζ). By the previous lemma this implies that
X(X(ϕ)) > −|X|2 .
Now we go on to proof the uniform estimates for X(ϕ) which is a real valued function by assumption.
The estimate for X(ϕ) follows from the fact that ϕ decays very fast as x → ∞ while (as we know form
the lemma above) X(X(ϕ)) is bounded from below by function that grows much slower compared to
exp(θX ). The bounds on the first derivative are then achieved via interpolation.
Lemma 5.4.14. Let ϕ ∈ U 3,α be a solution of (5.5). Then there exists a uniform constant C3 such that
kX(ϕ)kC 0 ≤ C3 .
Proof. Let x ∈ M . Let γx (t) := Φζt (x) (where Φζt is the flow of the (complete) vector field ζ),
t ∈ (−∞, ∞), ϕx (t) := ϕ(Φζt (x)).
(1) First we show that X(ϕ) is bounded from above: Let > 0 to be fixed later and α : [0, ∞) → R
such that
• 0 ≤ α(t) ≤ 1, t ∈ [0, ∞),
• α(0) = 1, α(t) = 0, t > ,
• α0 (0) = 0 = α0 (),
• sup |α0 (t)| < 2/, sup |α00 (t)| < 2/2 .
54
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
Then
Z
−
∞
|α00 (t)|C(x)dt ≤ −
0
Z
α00 (t)ϕx (t) =
Z
α0 (t)ϕ0x (t)dt
Z = −ϕ0x (0) −
α(t)ϕ00x (t)dt
0
Z 0
α(t)X(X(ϕ))(γx (t))dt
= −ϕx (0) −
Z0 ≤ −ϕ0x (0) +
|X|2 (γx (t))dt
0
0
0
where C(x) := supt∈[0,∞) |ϕx (t)|. And so
Z Z 2C(x)
ϕ0x (0) ≤
+ C1 (eλ − 1)|X|2 (x),
C(x)|α00 (t)| +
|X|2 (γx (t))dt ≤
0
0
where we used that
|X|2 (γ(t)) ≤ eλt |X|2 (γx (0)),
where
λ := 2 sup |∇X|.
M
This follows from
d
|X|2 (γx (t)) ≤ 2|∇X|(γx (t))|X|2 (γx (t)) ≤ 2λ|X|2 (x).
dt
Setting
1
1
:= log 1 + 1+|X|
2 (x)
λ
it follows that
C(x)λ
+ 1.
X(ϕ)(x) ≤
1
log 1 + 1+|X|
2 (x)
But since C(x)λ(log(1 + (1 + |X|2 (x))−1 ))−1 can be uniformly bounded by the previous observations we have a uniform upper bound C30 for X(ϕ). Indeed, we know that ζ is just the
radial vector field on the set Ωκ for some sufficiently large κ. And on the complement of Ωκ
we have |X|2 < κ which gives along with the uniform bound for C(x) on M a uniform upper
bound for X(ϕ) on Ωcκ , namely
sup X(ϕ) ≤
Ωcκ
C1
.
log(1 + κ−1 )
In Ωκ we know that
ϕ(Φζt (x)) ≤ C2 exp(−θX (Φζt (x))) = C2 exp(−θX (x)),
since
So (using |X|2 = θX
d
θX (Φζt (x)) = hζ, ∇θX i = |ζ|2 ≥ 0.
dt
on Ωκ )
sup X(ϕ)(x) ≤
x∈Ωκ
C2 exp(−θX (x))
C2 exp(−y)
≤ sup
≤ C30 .
1
log(1
+ 1/(1 + y))
y>κ
log 1 + 1+|X|2 (x)
(2) Secondly we establish a lower bound C300 for X(ϕ). Let β : (−∞, 0] → R be a function such
that
• 0 ≤ β(t) ≤ 1, t ∈ (−∞, 0],
• β(0) = 1, β 0 (0) = 0,
• β(t) = 0, t ≤ −, β 0 (0) = β 0 (−) = 0,
• sup |β 0 (t)| ≤ 2/, sup |β 00 (t)| ≤ 2/2 .
5.4. PROOF OF THE EXISTENCE THEOREM
55
Then
ϕ0x (0) =
Z
0
−
d
(β(t)ϕ0x (t)) =
dt
Z
0
β 0 (t)ϕ0x (t) + β(t)ϕ00x (t)dt
−
Z 0
≥
−β 00 (t)ϕx (t) − β(t)|X|2 (γx (t))
−
2
≥ − C̃(x) − (eλ − 1)|X|2 (x),
where C̃(x) := supt∈(−,0] |ϕx (t)| and we use that
|X|2 (γx (t)) ≤ (eλ|t| − 1)|X|2 (γx (0)).
So choosing
:= min
1
log 1 +
λ
1
1+|X|2 (x)
1
,
2
we get
2λC̃(x)
X(ϕ)(x) ≥ −
log 1 +
1
1+|X|2 (x)
−1
which gives a uniform lower bound for X(ϕ). Indeed,
C̃(x) =
sup |ϕ(t)| ≤ C1 exp(−
t∈(−,0]
inf
t∈(−,0]
θX (γx (t))) ≤ C1 exp(−(1 − )θX (x) + C)
All together we have
|X(ϕ)| ≤ C3 .
5.4.1.5. Calculations at a point. Before we continue with the higher order estimates we recall the
existence of holomorphic normal coordinates at a point with respect to a given Kähler metric g
Lemma 5.4.15. [Joy00] Let (M, J, g) be as above. Then there exists a positive number r > 0 and C > 0
such that at each point p ∈ M there exist an open neighborhood U of p and holomorphic coordinates
z1 , ..., zn : U → B(0, 2r) ⊂ Cn such that gi̄ (p) = δi̄ , C −1 δ ≤ (gi̄ ) ≤ Cδ in Br (0), and Γkij (p) = 0.
Moreover, if g 0 is the metric associated to ωϕ , ϕ ∈ U 3,α then the coordinates the coordinates can be
chosen such that they satisfy in addition gi̄ = (1 + ϕi̄ (p))δi̄ .
It follows that in holomorphic normal coordinates we have at p
ω n (p)
ωϕ (p)n
= dz1 ∧ dz̄1 ∧ ...dzn ∧ dz̄n ,
n
Y
=
(1 + ϕiı̄ (p))ω(p)n ,
i=1
gϕi̄
=
∆ϕ(p)
=
1
, and
1 + ϕiı̄ (p)
n
X
(1 + ϕiı̄ (p)) − n.
i=1
5.4.1.6. Second order estimates estimates. Now we can return to Yau’s and Aubin’s original proof
whose heart are the so-called C 2 -estimates. The next lemma is proven almost exactly like in Aubin’s
and Yau’s previous work. The only difference are the additional terms involving X.
Lemma 5.4.16. Let ϕ ∈ U 3,α be a solution of (5.5) then we can find a uniform constant C4 such that
kn + ∆ϕkC 0 ≤ C4 .
56
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
Proof. We follow [Tia00]. Consider
n
ωϕ
log
+ X(ϕ) − ϕ = F.
ωn
Differentiating by ∂zk we get
gϕi̄
∂gi̄
∂3ϕ
+
∂zk
∂zk ∂zi ∂ z̄̄
− g i̄
∂gi̄
∂ F̃
=
,
∂zk
∂zk
where
F̃ = F − X(ϕ) + ϕ.
And differentiating again by ∂z̄l̄ :
2
∂ 2 F̃
∂ gi̄
∂4ϕ
∂gts̄ ∂gi̄
∂ 2 gi̄
= gϕi̄
+
+ g t̄ g is̄
− g i̄
∂ z̄l̄ ∂zk
∂ z̄l̄ ∂zk
∂ z̄l̄ ∂zk ∂zi ∂ z̄̄
∂ z̄l̄ ∂zk
∂zl̄ ∂zk
3
3
∂ ϕ
∂gi̄
∂ ϕ
∂gts̄
+
+
.
− gϕt̄ gϕis̄
∂ z̄l̄
∂ z̄l̄ ∂zt z̄s̄
∂zk
∂zk ∂zi ∂ z̄̄
So in normal coordinates at a point p with respect to g we have after taking the trace (in k, ¯l) with
respect to g:
2
∂ 2 gi̄
∂3ϕ
∂3ϕ
∂4ϕ
kl̄ t̄ is̄
kl̄ i̄ ∂ gi̄
i̄ kl̄
+
− g gϕ gϕ
−g g
.
∆F̃ = gϕ g
∂ z̄l̄ ∂zk ∂zi ∂ z̄̄
∂ z̄l̄ ∂zk
∂ z̄l̄ ∂zk
∂ z̄l̄ ∂zt ∂ z̄s̄
∂zk ∂zi ∂ z̄̄
Using that
∆0 ∆ϕ = gϕkl̄
∂2
∂ z̄l̄ ∂k
= gϕkl̄ g i̄
∂2ϕ
g i̄
∂zi ∂ z̄̄
∂4ϕ
∂ 2 g i̄ ∂ 2 ϕ
+ gϕkl̄
∂ z̄l̄∂zk ∂zi ∂z̄
∂ z̄l ∂zk ∂zi ∂ z̄̄
we get
∂ 2 gi̄
∂ 2 g i̄ ∂ 2 ϕ
∂ 2 gi̄
− g kl̄ g i̄
∆F̃ = ∆0 ∆ϕ − gϕkl̄
+ gϕi̄ g kl̄
∂ z̄l̄ ∂zk ∂zi ∂ z̄̄
∂ z̄l̄ ∂zk
∂ z̄l̄ ∂zk
3
3
∂ ϕ
∂ ϕ
− g kl̄ gϕt̄ gϕis̄
.
∂ z̄l̄ ∂zt ∂ z̄s̄
∂zk ∂zi ∂ z̄̄
Now
∂ 2 gi̄
= −Ri̄kl̄
∂ z̄l̄ ∂zk
and
∂ 2 g i̄
= g in̄ g m̄ Rmn̄kl̄ .
∂ z̄l̄ ∂zk
So
∆F̃ = ∆0 ∆ϕ − gϕkl̄ Rij̄kl̄
∂2ϕ
− gϕi̄ g kl̄ Ri̄kl̄ + g kl̄ g i̄ Ri̄kl̄ − g kl̄ gϕt̄ gϕis̄
∂zi ∂ z̄̄
∂3ϕ
∂ z̄l̄ ∂zt ∂ z̄s̄
∂3ϕ
∂zk ∂zi ∂ z̄̄
. (5.7)
We can re-choose the coordinates such that both gi̄ = δi̄ and (gϕ )i̄ = (1 + ϕi̄ )δi̄ at the fixed point
p ∈ M . So the above equation simplifies to
1
1
1
1
R ϕiı̄ −
R
+ Riı̄kk̄ −
ϕ ϕ̄ik .
∆F̃ = ∆0 ∆ϕ −
1 + ϕkk̄ iı̄kk̄
1 + ϕiı̄ iı̄kk̄
1 + ϕiı̄ 1 + ϕj̄ ı̄j k̄
Letting C = Rmin := inf i6=k Riı̄kk̄ we get
1
ϕiı̄
1
(ϕkk̄ − ϕiı̄ )2
Riı̄kk̄ −1 +
+
= Riı̄kk̄
1 + ϕiı̄
1 + ϕkk̄
2
(1 + ϕiı̄ )(1 + ϕkk̄ )
C ((1 + ϕkk̄ ) − (1 + ϕiı̄ ))2
≥
2
(1 + ϕiı̄ )(1 + ϕkk̄ )
1 + ϕiı̄
=C
−1 .
1 + ϕkk̄
5.4. PROOF OF THE EXISTENCE THEOREM
57
So
∆0 (∆ϕ) ≥
1
1
ϕi̄k ϕı̄j k̄ + ∆F̃ + C (n + ∆ϕ)
− n2 .
(1 + ϕiı̄ )(1 + ϕkk̄ )
1 + ϕiı̄
Now let u := e−λϕ (n + ∆ϕ) where λ is to be determined later. Then
∆0 u = e−λϕ ∆0 ∆ϕ + 2h∇0 e−λϕ , ∇0 (n + ∆ϕ)igϕ + ∆0 e−λϕ (n + ∆ϕ)
= e−λϕ ∆0 ∆ϕ − 2λe−λϕ h∇0 ϕ, ∇0 ∆ϕigϕ − λe−λϕ ∆0 ϕ(n + ∆ϕ) + λ2 |∇0 ϕ|2gϕ e−λϕ (n + ∆ϕ)
≥ e−λϕ ∆0 ∆ϕ − e−λϕ gϕiı̄ (n + ∆ϕ)−1 (∆ϕ)i (∆ϕ)ı̄ − λe−λϕ ∆0 ϕ(n + ∆ϕ),
using the Schwarz inequality. Finally (again using the Cauchy-Schwarz inequality)
gϕiı̄ (n + ∆ϕ)−1 (∆ϕ)i (∆ϕ)ı̄
1
|ϕ |2
= (n + ∆ϕ)−1
1 + ϕiı̄ kk̄i
2
X
1
−1/2
1/2 −1
ϕkk̄i (1 + ϕkk̄ )
(1 + ϕkk̄ ) = (n + ∆ϕ)
1 + ϕiı̄ k
X
X
1
≤ (n + ∆ϕ)−1
|ϕkk̄i |2 (1 + ϕkk̄ )−1
(1 + ϕll̄ )
1 + ϕiı̄
k
l
X
1
1
ϕ ϕ
=
1 + ϕiı̄
1 + ϕkk̄ kk̄i k̄kı̄
k
X
1
1
=
ϕ ϕ
1 + ϕiı̄
1 + ϕkk̄ k̄ik kı̄k̄
k
1
1
ϕ̄ik ϕjı̄k̄ .
≤
1 + ϕiı̄ 1 + ϕkk̄
Hence
1
∆0 (e−λϕ (n + ∆ϕ)) ≥ e−λϕ ∆F̃ + C(n + ∆ϕ)
− n2 − λe−λϕ ∆0 ϕ(n + ∆ϕ)
1 + ϕiı̄
1
− n2 − λe−λϕ ∆0 ϕ(n + ∆ϕ).
≥ e−λϕ ∆F + ∆ϕ − ∆X(ϕ) + C(n + ∆ϕ)
1 + ϕiı̄
Now
∆X(ϕ) = ∇i X k ϕı̄k + X k ∇i ∇ı̄ ∇k ϕ
= ∇i X k ϕı̄k + X k ∇k ∇i ∇ı̄ ϕ
= ∇i X k ϕı̄k + X(n + ∆ϕ)
= ∇i X k (δı̄k + ϕı̄k − δı̄k ) + X(n + ∆ϕ)
≤ C 0 (n + ∆ϕ) − ∇i X k δkı̄ + X(n + ∆ϕ)
where we use |∇X| ≤ C 0 and
e−λϕ ∆X(ϕ) ≤ X(e−λϕ (n + ∆ϕ)) + C 0 (n + ∆ϕ)e−λϕ + C 0 ,
where we use the uniform upper bounds for kϕkC 0 (M ) and kX(ϕ)kC 0 (M ) .
It follows that
1
0
−λϕ
− C 0 − X(u) − C 0 u + u − λ∆0 ϕu.
∆u≥e
∆F + C(n + ∆ϕ)
1 + ϕiı̄
Setting λ := 1 − C ≥ 0 (where we can assume without loss of generality that C ≤ 1) and using
X
1
∆0 ϕ = n −
1 + ϕiı̄
i
58
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
we get for some uniform constants c0 , c2
∆0 u ≥ −c0 − X(u) − c2 u + e−λϕ
X
i
1
(n + ∆ϕ).
1 + ϕiı̄
But
X
1
≥
1 + ϕiı̄
P
1
1
1
(1 + ϕiı̄ ) (n−1)
Q
= e− n−1 (F −X(ϕ)+ϕ) (n + ∆ϕ) n−1 ,
(1 + ϕiı̄ )
so
e−λϕ
X
i
λϕ
1
1
1
(n + ∆ϕ) ≥ e− n−1 (F −X(ϕ)+ϕ)− n−1 u (n−1)
1 + ϕiı̄
which implies (using the estimates of X(ϕ) and ϕ) for some uniform constants c1 and c3
n
∆0 u ≥ −c1 − c2 u − X(u) + c3 u n−1 .
So by the maximum principle it follows that u and so |∆ϕ| is uniformly bounded.
Corollary 5.4.17. Let ϕ ∈ U 3,α be a solution of (5.5) then there exists a uniform constant C5 = C5 (g)
¯ C 0 (M ) ≤ C5 .
such that k∂ ∂ϕk
√
¯ > −ω. And so
Proof. Since ωϕ is positive we know that −1∂ ∂ϕ
√
¯ 2 ≤ (n + ∆ϕ)2 ≤ C 2 .
kω + −1∂ ∂ϕk
4
5.4.1.7. Third order estimates. We go on to prove the so-called C 3 -estimates. Of course we do not
need the full C 3 -estimates of ϕ. What we need to start the Schauder machine is simply C 0,α -estimates
¯ However, even though Yau’s calculation (which we reproduce here) is quite long and tedious it
for ∂ ∂ϕ.
is self-contained and uses, apart from the maximum principle, only algebraic identities. An alternative
¯ (or
way to prove C 2,α -estimates of ϕ (avoiding any Schauder) directly from the C 0 -estimates for ∂ ∂ϕ
∆ϕ) is to use results obtained after Yau’s proof of the Calabi conjecture by Evans, Krylov and Safonov
among others. Namely the C 0 -estimates for ∆ϕ and ϕ give us (via Morrey) a uniform C 1,α estimate for
ϕ, α ∈ (0, 1). It follows from Krylov theory using the uniform ellipticity and concavity of the operator
in question that there exists a uniform bound for the C 2,γ -norm of ϕ where γ ∈ (0, α). More precisely,
the following theorem is used:
Theorem 5.4.18. [CW98] Assume that Ω ⊂ Rn has a smooth boundary, and the boundary is smooth.
n
n
are
(where S+
Assume G(x, u, Du, D2 u) is smooth in all variables (x, z, p, r) ∈ Γ = Ω × R × Rn × S+
2
the positive symmetric matrices), uniformly elliptic and concave (or convex) in D u, and assume that
kukC 1,α (Ω̄) is bounded for some 0 < α ≤ 1. Then there are constants 0 < β < α and C so that for any
0 < γ < β, and Ω0 ⊂ Ω we have for any solution u of G(x, u, Du, D2 u) = 0
kukC 2,γ (Ω̄0 ) ≤ C.
where β depends on kD2 ukC 0 (Ω) , kukC 1,α (Ω) , and kGkC 1,α Ω , the ellipticity Λ/λ of G and C depends on
dist(Ω0 , ∂Ω).
Remark 5.4.19. By a bound C for kGkC 1,α we mean that G = G(x, z, p, r) satisfies
(1) supΓ (|∂x G| + |∂z G| + |∂p G|) < C(1 +
|r|)
(2) supΓ [∂r G]α,x + [∂r G]α,p + [∂z G]α,z < C
(3) supΓ [∂p G]α,r + [∂z G]α,r + [∂x G]α,r < C and
(4) supΓ [∂p G]α,p + [∂p G]α,z + [∂p G]α,x + [∂z G]α,z + [∂z G]α,x + [∂x G]α,x < C(1 + |r|),
where the notation [ · ]α,y just refers to the Hölder seminorm in the corresponding variable y.
5.4. PROOF OF THE EXISTENCE THEOREM
59
In our case
G(x, u, Du, D2 u) = log
ωun
ωn
− u + X(u) − F (x),
i.e.
G(x, z, p, r) = log
(ω + π(r))n
ωn
+ hX, pi − z − F (x),
where π maps r to 21 (r + JrJ t ). Unfortunately we do not have uniform bound for k∂p GkC 0,α (Ω) . Namely
we have
k∂p GkC 0,α (Ω) = kXkC 0,α (Ω)
which is not bounded as Ω moves towards infinity. This means that we either have to find a good decay
estimate for X(ϕ) first or show that the condition on G can be relaxed. The second method would be
to go through the details of Evans’ and Krylov’s proof which would probably take much longer than the
C 3 -estimates of Yau which can be applied without any major changes.
Lemma 5.4.20. Let ϕ ∈ U 3,α be a solution of (5.5). Then there exists a uniform constant C6 such that
¯ C 0 (M ) ≤ C6 .
k∇∂ ∂ϕk
Proof. The proof works just like in [Yau78]. We will omit some of the point-wise calculations
since they are almost the same as in [Yau78]. Set
¯ 2 .
S 2 := |∇∂ ∂ϕ|
gϕ
Then the following expression can be found in [Yau78]:
∆0 S 2 ∼
=
X
(1 + ϕaā )−1 (1 + ϕiı̄ )−1 (1 + ϕj̄ )−1 1 + ϕkk̄ )−1

2

X
−1 × ϕı̄j k̄a −
ϕı̄pk̄ ϕp̄ja (1 + ϕpp̄ ) 
p
2 

X
+ ϕi̄ka −
(ϕp̄ia ϕp̄k + ϕpı̄k ϕp̄a )(1 + ϕpp̄ )−1 
p
+ 2< (1 + ϕiı̄ )−1 (1 + ϕj̄ )−1 (1 + ϕkk̄ )−1 ϕı̄j k̄ F̃i̄k
− 2(1 + ϕiı̄ )−1 (1 + ϕj̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕrr̄ )−1 F̃rı̄ ϕi̄k ϕr̄j k̄
− (1 + ϕiı̄ )−1 (1 + ϕj̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕrr̄ )−1 F̃j r̄ ϕi̄k ϕı̄rk̄
where
F̃ = F + ϕ − X(ϕ)
and A ∼
= B means that there exist constants B1 B2 , and B3 such that
|A − B| ≤ B1 S 2 + B2 S + B3 .
60
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
We have one additional term:
¯ 2
X(S 2 ) = ∇0 |∇∂ ∂ϕ|
X
=
gϕ
gϕi̄ gϕkl̄ gϕmn̄ X p ∇0p (ϕil̄m ϕ̄kn̄ )
= gϕi̄ gϕkl̄ gϕmn̄ X p ϕil̄mp ϕ̄kn̄ + ϕil̄m ϕ̄kn̄p − (Γ0 )qpi ϕql̄m ϕ̄kn̄ − (Γ0 )qpm ϕil̄q ϕ̄kn̄ − (Γ0 )qpk ϕil̄m ϕ̄qn̄
= gϕi̄ gϕkl̄ gϕmn̄ X p ϕil̄mp ϕ̄kn̄ + ϕil̄m ϕ̄kn̄p − gϕqt̄ ϕit̄p ϕql̄m ϕ̄kn̄ − gϕqt̄ ϕmt̄p ϕil̄q ϕ̄kn̄ − gϕqt̄ ϕkt̄p ϕil̄m ϕ̄qn̄
= (1 + ϕiı̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕmm̄ )−1 X p ϕik̄mp ϕı̄km̄ + ϕik̄m ϕı̄km̄p
− (1 + ϕiı̄ )−1 (1 + ϕkk̄ )−1
× (1 + ϕmm̄ )−1 (1 + ϕqq̄ )−1 X p ϕiq̄p ϕqk̄m ϕı̄km̄ + ϕmq̄p ϕik̄q ϕı̄km̄ + ϕkq̄p ϕik̄m ϕı̄qm̄
∼
= (1 + ϕiı̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕmm̄ )−1 (X(ϕ)ik̄m ϕı̄km̄ + ϕik̄m X(ϕ)ı̄km̄ )
− (1 + ϕiı̄ )−1
× (1 + ϕkk̄ )−1 (1 + ϕmm̄ )−1 (1 + ϕqq̄ )−1
× X(ϕ)iq̄ ϕqk̄m ϕı̄km̄ + X(ϕ)mq̄ ϕik̄q ϕı̄km̄ + X(ϕ)kq̄ ϕik̄m ϕı̄qm̄
= (1 + ϕiı̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕmm̄ )−1 2< (X(ϕ)ik̄m ϕı̄km̄ )
− (1 + ϕiı̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕmm̄ )−1 (1 + ϕqq̄ )−1
× X(ϕ)iq̄ ϕqk̄m ϕı̄km̄ + X(ϕ)mq̄ ϕik̄q ϕı̄km̄ + X(ϕ)kq̄ ϕik̄m ϕı̄qm̄
∼
= −(1 + ϕiı̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕmm̄ )−1 2< F̃ik̄m ϕı̄km̄
+ (1 + ϕiı̄ )−1 (1 + ϕkk̄ )−1 (1 + ϕmm̄ )−1 (1 + ϕqq̄ )−1 F̃iq̄ ϕqk̄m ϕı̄km̄ + F̃mq̄ ϕik̄q ϕı̄km̄ + F̃kq̄ ϕik̄m ϕı̄qm̄
So
∆0 S 2 + X(S 2 ) ∼
=
X
(1 + ϕaā )−1 (1 + ϕiı̄ )−1 (1 + ϕj̄ )−1 (1 + ϕkk̄ )−1

2

X
× ϕı̄j k̄a −
ϕı̄pk̄ ϕp̄ja (1 + ϕpp̄ )−1 
p
2 

X
+ ϕi̄ka −
(ϕp̄ia ϕp̄k + ϕpı̄k ϕp̄a )(1 + ϕpp̄ )−1 
p
On the other hand (by the previous estimates)
∆0 (∆ϕ) + X(∆ϕ) ≥ (1 + ϕkk̄ )−1 (1 + ϕll̄ )−1 |ϕkı̄̄ |2 − C
for some uniform constant C. So we can find a uniform constant C 0 such that
∆0 (S 2 + C 0 ∆ϕ) + X(S 2 + C 0 ∆ϕ) ≥ C 00 S 2 − C 000
For some positive constants C 00 > 0 and C 000 > 0. By the maximum principle we obtain a uniform
estimate for S.
This finishes the proof of theorem 5.4.1. Indeed setting D2 = C2 , D3 = C3 , D4 = C5 , and D5 = C6 we
get
¯ C 0 (M ) ≤ D4 , and k∇∂ ∂ϕk
¯ C 0 (M ) ≤ D5 .
kϕkC 0 (M ) ≤ D2 , kX(ϕ)kC 0 (M ) ≤ D3 , k∂ ∂ϕk
w
Proof of theorem 5.4.2. We now use theorem 5.4.1 to show 5.4.2. First we make the following
simple observation:
5.4. PROOF OF THE EXISTENCE THEOREM
61
Lemma 5.4.21. Let ϕ ∈ U 3,α be a solution of (5.5). Then for any α ∈ (0, 1) there exists a uniform
constant D9 such that
kgϕ−1 gkC 0,α (M ) ≤ D9 .
Proof. Let hϕ := g −1 gϕ then khϕ kC 0,α (M ) ≤ D80 by theorem 5.4.1. In particular the positive
eigenvalues λi (h) of hϕ are uniformly bounded. Now since hϕ also solves
det(hϕ ) = exp(ϕ − X(ϕ) + F )
and ϕ and Q
X(ϕ) are uniformly bounded it follows that there exists a uniform positive constant c > 0
n
such that i=1 λi > c. But this implies that each λi is uniformly bounded from below and hence
−1
khϕ kC 0 (M ) ≤ C for some uniform constant. The estimates of kh−1
ϕ kC 0,α (M ) can be obtained similarly.
2,α
5.4.1.8. Cw
-estimates. We are now finally ready to prove that a solution ϕ ∈ U 3,α of (5.5) is
2,α
uniformly bounded in the Cw
-norm. The first step is to show that for any µ ∈ (0, 1) we can find a
constant C7 = C7 (g) such that kX(ϕ) exp(µθX )kC 0,α (M ) ≤ C7 . The function (X(ϕ) − ϕ) satisfies an
elliptic equation and the maximum principle gives the desired estimate. In a second step we show that
for any µ ∈ (0, 1) there exists a constant C70 such that kϕ exp(µθX )kC 1,α (M ) < C70 . This is done using the
local Morrey-Schauder estimates D.2.1, theorem 5.4.1 and the previously obtained estimates for X(ϕ).
And again using Schauder estimates D.2.1 we show that for any µ we can find a constant C700 such that
kϕ exp(µθX )kC 2,α (M ) < C700 . The last step is to apply the global Schauder estimates in appendix A to
the linear elliptic partial differential equation (the linear part of the Taylor expansion of the nonlinear
Monge Ampere operator) whose right hand (the quadratic remainder term of the expansion) decays
2,α
-estimates for ϕ.
sufficiently fast. This gives the uniform Cw
We start with three lemmas that will help us to set up the equation that X(ϕ) satisfies and to get
estimates for the coefficients in this equation.
Lemma 5.4.22. Let ϕ be a solution of (5.5) then outside of a compact set it satisfies
∆0 (X(ϕ) − ϕ) + X(X(ϕ) − ϕ) = 0,
where ∆0 = ∆gϕ
√
¯ X and F = 0. So
Proof. Note that outside of a compact set ω = −1∂ ∂θ
n
ωϕ
X(−X(ϕ) + ϕ) = ∇X log
ωn
= gϕi̄ ∇X ∇i ∇̄ ϕ
= gϕi̄ X k ∇k ∇i ∇̄ ϕ
= ∆0 X(ϕ) − gϕi̄ g kl̄ ∇i ∇l̄ θX (∇k ∇̄ ϕ)
2 2
∂ ϕ
0
i̄
kl̄ ∂ θX
= ∆ X(ϕ) − gϕ g
∂zi ∂zl̄
∂zk ∂z̄
∂2ϕ = ∆0 X(ϕ) − gϕi̄ g kl̄ gil̄
∂zk , ∂z̄
2 ∂
ϕ
= ∆0 X(ϕ) − gϕi̄ δik
∂zk ∂z̄
2 ∂ ϕ
= ∆0 X(ϕ) − gϕk̄
∂zk ∂z̄
= ∆0 X(ϕ) − ∆0 ϕ,
where we used the fact that ∇̄ X = 0 and ∇X ϕ = ∇<(X) ϕ. It follows that
∆0 (X(ϕ) − ϕ) = X(ϕ − X(ϕ))
62
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
Lemma 5.4.23. If θX is a potential for X with respect to g then θX + X(ϕ) is a potential for X with
respect to gϕ .
Proof. Note first that
Hence
ıX ωϕ = ıX ω +
√
√
¯ X.
ıX ω = X i −1gi̄ dz̄ ̄ = −1∂θ
√
¯ =
−1ıX ∂ ∂ϕ
√
¯ X+
−1∂θ
√
¯ X ∂ϕ =
−1∂ı
√
¯ X + X(ϕ))
−1∂(θ
in other words
(1,0)
grad(1,0)
θX = X
gϕ (θX + X(ϕ)) = gradg
Lemma 5.4.24. Let ϕ be a solution of (5.5). Then ∆0 (θX + X(ϕ)) grows at most linearly.
¯ and hence g −1 we know that ∆0 θX is uniformly bounded. As for
Proof. By the estimates for ∂ ∂ϕ
ϕ
the remaining term
2
¯ g + |g −1 ||∇∇∇ϕ|
¯
∆0 X(ϕ) ≤ |g −1 |g |∇X|g |∇∇ϕ|
|X|g ,
ϕ
ϕ
g
which proves the lemma.
Lemma 5.4.25. Let ϕ be a solution of (5.5). Then for any µ ∈ (0, 1) there exists a uniform constant
C7 (µ, g) such that |X(ϕ)| ≤ C7 exp(−µθX ).
Proof. Let µ ∈ (0, 1) and ψ = (X(ϕ) − ϕ)eµ(θX +X(ϕ)) . Then outside of a compact set:
∆0 ψ − (2µ − 1)X(ψ) − µ∆0 (θX + X(ϕ)) + (µ − µ2 )|X|2gϕ ψ = 0.
Indeed using lemma 5.4.1.8,
∆0 ψ = eµ(θX +X(ϕ)) ∆(X(ϕ) − ϕ) + 2µX(X(ϕ) − ϕ)eµ(θX +X(ϕ)) + (µ∆0 (θX + X(ϕ)) + µ2 |X|2gϕ )ψ
= (−1 + 2µ)X(X(ϕ) − ϕ)eµ(θX +X(ϕ)) + (µ∆0 (θX + X(ϕ)) + µ2 |X|2gϕ )ψ
= (−1 + 2µ)X(ψ) − µ(2µ − 1)|X|2gϕ ψ + (µ∆0 (θX + X(ϕ)) + µ2 |X|2gϕ )ψ
= (−1 + 2µ)X(ψ) + (µ∆0 (θX + X(ϕ)) + (µ − µ2 )|X|2gϕ )ψ.
Now, since µ < 1 we have µ − µ2 > 0 and by the previous estimates the term (µ − µ2 )|X|2gϕ dominates
∆0 (θX + X(ϕ)) and so there is a compact set Kµ which is uniformly bounded such that on M \ Kµ :
µ∆0 (θX + X(ϕ)) + (µ − µ2 )|X|2gϕ ≥ 1.
It follows from the maximum/minimum principle that
sup |ψ| ≤ sup |ψ|,
M
Kµ
and using that ϕ is exponentially decaying and the fact that |X(ϕ)| is uniformly bounded we can find
a constant C7 (µ, g) such that
|X(ϕ)| ≤ C7 e−µθX .
Throughout the rest of this section we will make use of the following lemma:
Lemma 5.4.26. Suppose for some r > 0 and x ∈ M and µ ∈ (0, 1) there exists a constant C = C(µ)
such that kukC k,α (Br (x)) ≤ C supB2r (x) exp(−µθX ) then for any µ ∈ (0, 1) there exists a uniform constant
C = C(r, µ, g) such that k exp(µθX )ukC k,α (M ) ≤ C.
5.4. PROOF OF THE EXISTENCE THEOREM
63
Proof. Let µ ∈ (0, 1). By assumption we can pick µ0 ∈ (µ, 1) and P
a constant C = C(µ0 ) such
0
that kukC k,α (Br (x)) ≤ C supB2r (x) exp(−µ θX ). But then it follows that j |∇j u|(x) + [∇k u]α (x) ≤
C exp(−µ0 inf B2r (x) θX ). Since for any inf Br (x) θX ≥ θX − r supBr (x) |X| that
X
|∇j u|(x) + [∇k u]α (x) ≤ C exp(−µ0 θX + µ0 r sup |X|)
B2r (x)
j
And since |∇k θX | is uniformly bounded
X
|∇j exp(µθX )u|(x) + [∇k exp(µθX )u]α (x) ≤ Cpk,α (|X|) exp((µ − µ0 )θX + µ0 r sup |X|),
Br (x)
j
where p(|X|) ≤ C(1 + |X|)k+α with a uniform C = C(k, g). The right hand side in the equation above
can be bounded by a uniform constant depending on k and µ and so the lemma follows.
Lemma 5.4.27. Let ϕ be a solution of (5.5). Then for every µ ∈ (0, 1) there exists a uniform constant
¯ ≤ C8 exp(−µθX ).
C8 = C8 (µ, g) such that |∂ ∂ϕ|
Proof. Let K be a compact set such that F = 0 on M \ K and let x ∈ M \ K such that
B(x, 2r) ⊂ M \ K. Choose holomorphic coordinates on B(x, 2r) then
n
ωϕ
+ X(ϕ) − ϕ
0 = log
ωn
Z 1
2
i̄
=
gϕ
d ∂z∂i ∂ϕz̄̄ + X(ϕ) − ϕ
0
2
= ai̄ ∂z∂i ∂ϕz̄̄ + X(ϕ) − ϕ.
By the estimates above kai̄ kC 0,α (B(x,2r)) ≤ Λ(g) and ai̄ ≥ λ(g)δ i̄ (This follows form the estimates for
¯
¯
And so by the local Morrey–Schauder
|∂ ∂ϕ|,
inf ωϕn /ω n , i.e. the estimates for X(ϕ) − ϕ, and |∇∂ ∂ϕ|).
estimates D.2.1
kϕkC 1,α (Br (x)) ≤ C(r, g) kϕkC 0 (B2r (x)) + kX(ϕ) − ϕkC 0 (B2r (x)) ,
where C is independent of x. It follows in particular that for every µ ∈ (0, 1) there exists a uniform
constant D(µ, r, g) such that
|X(ϕ) − ϕ| + [X(ϕ) − ϕ]α ≤ De−µθX .
And applying again the Schauder estimates D.2.1
kϕkC 2,α (Br (x)) ≤ C(Λ, r) kϕkC 0 (B2r (x)) + kX(ϕ) − ϕkC 0,α (B2r (x))
we can conclude that for any µ ∈ (0, 1) we can find a uniform constant C8 (µ, g) such that
¯ + ∂ ∂ϕ
¯
|∂ ∂ϕ|
≤ C8 e−µθX .
α
Lemma 5.4.28. Let ϕ ∈ U 3,α be a solution of (5.5). Then there exists a uniform constant C9 = C9 (g)
such that
kϕkCw2,α (M ) ≤ C9 .
Proof. For u ∈ K2,α let
M (u) := log
(ω +
√
¯ n
−1∂ ∂u)
ωn
+ X(u) − u.
Then
M (0) = 0
M 0 (0)v = ∆v + X(v) − v
¯ 2 ,
M 00 (u)(v, v) = gui̄ vil̄ gukl̄ vk̄ = |∂ ∂v|
gu
64
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
√
¯ So
where gu is the Kähler metric associated to ω + −1∂ ∂u.
√
¯ n
(ω + −1∂ ∂u)
MA(u, F ) := M (u) − F = log
+ X(u) − u − F
ωn
Z 1Z σ
M 00 (τ u)(u, u)dτ dσ
= MA(0, F ) + M 0 (0)u +
0
0
Z
= −F + ∆u + X(u) − u +
|
{z
}
1
Z
0
σ
0
¯ 2 dτ dσ.
|∂ ∂u|
gτ u
:=L(u)
And if ϕ is a solution of (5.5) then M A(ϕ, F ) = 0, and so
Z 1Z σ
¯ 2 dτ dσ
|∂ ∂ϕ|
L(ϕ) = F −
gτ ϕ
0
0
Note that by lemma 5.4.21 above gτ ϕ is uniformly equivalent to g for all τ ∈ [0, 1]. And since for any
µ ∈ (0, 1)
¯ ∈ C 1,α (M ) ⊂ C 0,α (M )
exp(µθX )∂ ∂ϕ
¯ is uniformly bounded in C 0,α (M ) by a constant depending only on
and for any µ ∈ (0, 1) exp(µθX )∂ ∂ϕ
2
¯
µ and g it follows that |∂ ∂ϕ| exp(2µθX ) ∈ C 0,α (M ) is uniformly bounded in C 0,α (M ). Hence (choosing
µ > 1/2)
Z 1Z σ
¯ 2 dτ dσ ∈ C 0,α (M ).
|∂ ∂ϕ|
gτ ϕ
w
0
0
0,α
is uniformly bounded in Cw
(M ). As seen above ϕ solves
Z 1Z σ
¯ 2 dτ dσ =: G.
L(ϕ) = F −
|∂ ∂ϕ|
gτ ϕ
0
0
0,α
(M ).
and by the observations above G is uniformly bounded in Cw
By assumption G ∈
Consider the conjugated operator
1,α
(M )
Cw
L̃ := exp(θX ) ◦ L ◦ exp(−θX )
acting on C
1,α
(M ). Then for u ∈ C
2,α
(M )
L̃(u) = ∆u − X(u) − (∆θX + 1)u.
Recall that ψ := exp(θX )ϕ ∈ C
2,α
(M ) (even though without uniform upper bound yet) and ψ satisfies
L̃(ψ) = ∆ψ − X(ψ) − (∆θX + 1)ψ
= exp(θX )L(ϕ)
= exp(θX )G.
Let
Ãu := ∆u − X(u) − (n + 1)u.
Then
L̃u − Ãu = (n − ∆θX )u,
where (n − ∆θX ) has compact support. And ψ satisfies
Ãψ = exp(θX )G + (n − ∆θX )ψ =: H
By the global Schauder estimates in the appendix, since ψ ∈ X̃ 3,α ⊂ X̃ 3,α ,
kψkC 2,α (M ) ≤ CkÃψkC 0,α (M )
for some uniform C = C(g). And so
kψkC 2,α (M ) ≤ C kGkCw0,α (M ) + k(n − ∆θX )ψkC 0,α (M )
= C(kG exp(θX )kC 0,α (M ) + k(n − ∆θX )ψkC 0,α (M ) ).
5.4. PROOF OF THE EXISTENCE THEOREM
65
And so it follows that
kϕkCw2,α (M ) ≤ C kGkCw0,α (M ) + k(n − ∆θX )kCw0,α (M ) kϕkC 0,α (M )
≤ C9 ,
for some uniform constant C9 = C9 (g).
3,α
3,α
5.4.1.9. Cw
-estimates. For the Cw
-estimates we proceed as follows. First from the local MorreySchauder estimates D.2.1 in chapter 2, the uniform C 0,α -estimates for gϕ−1 g and equation (5.7) we
2,α
get uniform estimates on kϕkC 3,α (M ) . The previously established Cw
-estimates imply that for any
µ ∈ (0, 1) we can find a bound C(µ, g) for k exp(µθX )(X(ϕ) − ϕ)kC 1,α (M ) . And so the local Schauder
estimates D.2.1 tell us that k exp(µθX )ϕkC 3,α (M ) has an upper bound depending only on µ and g.
1,α
Hence H from the proof of the previous theorem is uniformly bounded in Cw
. And finally by the
3,α
global Schauder estimates ϕ is uniformly bounded in Cw .
Lemma 5.4.29. Let ϕ ∈ U 3,α be a solution of (5.5). Then there exists a uniform constant C = C(g)
such that
kϕkC 3,α (M ) ≤ C.
Proof. By the second order estimates (equation (5.7)) we know that ϕ satisfies an equation of the
form
¯ ∗ g −1 + g −1 ∗ g −1 ∗ ∇∇∇ϕ
¯ ∗ ∇∇
¯ ∇ϕ.
¯
∆0 (∆ϕ + X(ϕ) − ϕ) = ∆F + Rm ∗ ∇∇ϕ
ϕ
ϕ
ϕ
We already have a uniform C 0 -estimate for the right hand side and kgϕ−1 kC 0,α (M ) is also uniformly
bounded. So by the Morrey-Schauder estimates D.2.1 there exists a uniform constant C = C(g) such
that
k∆ϕ + X(ϕ) − ϕkC 1,α (M ) ≤ C.
Moreover by the weighted estimates above we already know that X(ϕ) − ϕ are uniformly bounded in
C 1,α (M ). Hence by the local Schauder estimates D.2.1
kϕkC 3,α (M ) ≤ C.
The previous lemma implies
Corollary 5.4.30. There exists a uniform constant C such that kgϕ−1 gkC 1,α (M ) ≤ C.
Lemma 5.4.31. Let ϕ ∈ U 3,α be a solution of (5.5). Then for any µ ∈ (0, 1) there exists a uniform
constant C = C(g, µ) such that
k exp(µθX )ϕkC 3,α (M ) ≤ C
Proof. By the previous corollary we know that kgϕ−1 gkC 1,α (M ) is uniformly bounded. And so
arguing as in the lemma 5.4.27 (using the local Schauder estimates D.2.1) we can conclude that
k exp(µθX )ϕkC 3,α (M ) ≤ C for some constant depending on g and µ.
Corollary 5.4.32. Let G be as above. Then there exists a uniform constant C = C(g) such that
kGkCw1,α (M ) ≤ C.
Lemma 5.4.33. Let ϕ ∈ U 3,α be a solution of (5.5). Then there exists a uniform constant C10 such that
kϕkCw3,α (M ) ≤ C10 .
Proof. From the proof of the previous lemma we can conclude that ψ := exp(θX )ϕ ∈ X̃ 3,α satisfies
L̃ψ = ∆ψ − X(ψ) − (n + 1)ψ = H
for H ∈ C 1,α (M ) as before which is uniformly bounded in C 1,α (M ). It follows that |∇ψ|2 satisfies
¯
¯ 2 +Rc(∇ψ, ∇ψ)−h∇
¯
¯
∆|∇ψ|2 −X(|∇ψ|2 )−2(n+1)|∇ψ|2 = 2< h∇H, ∇ψi
+|∇∇ψ|2 +|∇∇ψ|
∇ψ X, ∇ψi =: K.
66
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
And since the right hand side is in C 0,α (M ) it follows by the global Schauder estimates that |∇ψ|2 ∈
C 2,α (M ) and hence ψ ∈ C 3,α (M ). Moreover there exists a uniform constant C(g) such that
kψkC 3,α (M ) ≤ C(kKkC 0,α (M ) + kψkC 1,α (M ) ).
3,α
And by the previous estimates we have ϕ ∈ Cw
(M ) and a uniform constant C10 = C10 (g) such that
kϕkC 3,α (M ) ≤ C10 .
5.4.1.10. X 3,α -estimates. It remains to show that X(exp(θX )ϕ) is uniformly bounded in C 1,α (M ).
Lemma 5.4.34. Let ϕ ∈ U 3,α be a solution of (5.5). Then there exists a uniform constant C11 such that
kX(exp(θX )ϕ)kC 1,α (M ) ≤ C11 .
Proof. Since ψ := exp(θX )ϕ solves
Ãψ = ∆ψ − X(ψ) − (n + 1)ψ = H
1,α
with H uniformly bounded in C (M ) and ∆ψ and ψ are uniformly bounded in C 1,α (M ) as well the
uniform estimate for X(ψ) follows immediately.
Combining lemmas 5.4.28, 5.4.33, 5.4.34 completes the proof of theorem 5.4.2.
Proof of theorem 5.4.3. Now we can obtain the higher order estimates with a bootstrapping
argument. Even though this is standard we will include a prove because it involves two steps unlike in
the compact case where it can be done directly.
Lemma 5.4.35. Let ϕ ∈ U 3,α be a solution of (5.5) and k ∈ N. Suppose that there exists a uniform
constant Ck = Ck (g) such that kϕkCwk+2,α (M ) ≤ Ck . Then there exists a uniform constant C = C(k, g)
such that kϕkC k+3,α ≤ C.
Proof. As before ϕ satisfies equation (5.7). Since gϕ−1 is bounded in C k,α (M ) and uniformly equivalent to g by assumption and the right hand side of equation (5.7) is uniformly bounded in C k−1,α (M )
it follows by the local Schauder estimates D.2.1 that there exists a uniform constant C = C(k, g) such
that k∆ϕ + X(ϕ) − ϕkC k+1,α (M ) ≤ C. By the assumption that kϕkCwk+2,α is uniformly bounded it follows X(ϕ) − ϕ is uniformly bounded in C k+1,α (M ). And so again by local Schauder estimates D.2.1 in
Chapter 2 there exists a uniform constant C = C(k, g) such that kϕkC k+3,α (M ) ≤ C.
Corollary 5.4.36. Let ϕ be as above. Then there exists a uniform constant C = C(k, g) such that
kgϕ−1 kC k+1,α (M ) ≤ C.
Lemma 5.4.37. Let ϕ ∈ U 3,α be a solution of (5.5) then for any µ there exists a constant C = C(g, k, µ)
such that k exp(µθX )ϕkC k+3,α (M ) ≤ C.
Proof. By the corollary above gϕ−1 is uniformly bounded in C k+1,α and so we can argue as in the
proof of lemma 5.4.27 that for any µ ∈ (0, 1) there exists a uniform constant C = C(k, g, µ) such that
k exp(µθX )ϕkC k+3,α (M ) ≤ C.
Lemma 5.4.38. Let ϕ ∈ U 3,α be a solution of (5.5) and ψ := exp(θX )ϕ. Let H := Ãψ then H ∈
C k+1,α (M ). Moreover, assume that there exists a uniform constant Ck0 = Ck0 (g) such that kϕkCwk+2,α (M ) ≤
0
Ck0 (g) then there exists a constant Ck+1 = Ck+1
(g) such that kHkC k+1,α (M ) ≤ Ck+1 .
RR
¯ 2 . By lemma 5.4.37 it
Proof. Recall that H = exp(θX )(G + (n − ∆θX )ϕ) and G = F −
|∂ ∂ϕ|
follows that there exists a uniformly constant C(k, g) such that kGkCwk+1,α (M ) . And so for some uniform
constant C = C(g, k) we have kHkC k+1,α ≤ C since (n − ∆θX ) has compact support.
Lemma 5.4.39. Let ϕ ∈ U 3,α be a solution of (5.5). Assume that there exists a uniform constant Ck
with k ∈ N0 such that
kϕkCwk+2,α ≤ Ck .
Then we can find a uniform constant Ck+1 = Ck+1 (g) such that
kϕkCwk+3,α ≤ Ck+1 .
5.4. PROOF OF THE EXISTENCE THEOREM
67
Proof. Recall that ψ := exp(θX ϕ) solves
Ãψ = ∆ψ − X(ψ) − (n + 1)ψ = H.
So |∇
k+1
2
ψ| satisfies
∆|∇k+1 ψ|2 + X(|∇k+1 ψ|2 ) − 2(n + 1)|∇k+1 ψ|2


k−3
X
¯ k+1 ψi + |∇∇
¯ k+1 ψ|2 + |∇k+2 ψ|2 + < 
¯ k+1 ψ 
= 2< h∇k+1 H, ∇
∇j Rm ∗ ∇k−1−j ψ ∗ ∇
j=0




k+1
k
X
X
¯ k+1 ψ  + < 
¯ k+1 ψ 
+ <
∇j X ∗ ∇k+2−j ψ ∗ ∇
∇j Rm ∗ ∇k−j ψ ∗ X ∗ ∇
j=1
j=0
=: Hk+1 .
By the lemma above and the assumptions Hk+1 ∈ C 0,α (M ). And by the global Schauder estimates it
follows that there exists a constant C = C(k, g) such that
k∇k+1 ψkC 2,α (M ) ≤ CkHkC 0,α (M )
and so we can find Ck+1 such that
kϕkCwk+3,α (M ) ≤ Ck+1 .
k+2,α
(M ) then X(eθX ϕ) ∈ C k,α (M ).
Lemma 5.4.40. Let ϕ be a solution in Cw
Proof. Let ψ := exp(θX )ϕ then
X(ψ) = ∆ψ − L̃(ψ) − (∆θX + 1)ψ
where that right hand side is uniformly bounded in C k,α (M ).
Proof of theorem 5.4.4. Our aim is to prove theorem 5.4.4 using the implicit function theorem.
So consider
n
ωϕ
+ X(ϕ) − ϕ − f.
MA(ϕ, f ) := (ϕ, f ) 7→ log
ωn
1,α
1,α
Lemma 5.4.41. The operator MA : U 3,α × Cw,X
(M ) → Cw,X
(M ) is well-defined.
Proof. We can write (see proof of lemma 5.4.28)
Z 1Z σ
¯ 2 dτ dσ + X(ϕ) − ϕ − f.
MA(ϕ, f ) = ∆ϕ −
|∂ ∂ϕ|
ωτ ϕ
0
0
1,α
Then by assumption the combination of the first and the last three terms is in Cw
(M ) (see the remark
1,α
k,α
above). The second term is in Cw (M ) since ωτ ϕ is equivalent to ω. More precisely, since ϕ ∈ Cw
(M )
2
−θ
k−2,α
¯ ∈ |X| e X C
the function |∂ ∂ϕ|
(M ) and so
¯ 2 ∈ |X|4 e−2θX C k−2,α ⊂ C k−2,α (M ).
|∂ ∂ϕ|
w
Note that if ϕ̃ ∈ U 3,α is a solution of (5.4) then MA(ϕ̃, F̃ ) = 0. To prove the theorem we have to show
1,α
that for any F ∈ Cc∞ (M ) sufficiently close to F̃ in the Cw
-norm satisfying Jζ(F ) = 0 we can find a
3,α
ϕ∈U
such that MA(ϕ, F ) = 0. It suffices to show that for a solution ϕ̃ ∈ U 3,α of (5.6)
1,α
Lϕ̃ : X 3,α → Cw,X
(M ),
ψ 7→
d
MA(ϕ̃ + ψ, F̃ )|=0 = ∆gϕ̃ ψ + X(ψ) − ψ
d
68
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
is an isomorphism. For notational convenience we will write L instead of Lϕ̃ . We define the operator
L̃ = L̃ϕ̃ formally by
L̃ϕ̃ ψ̃ := eθX +X(ϕ̃) L(e−θX −X(ϕ̃) ψ̃) = ∆gϕ̃ ψ − X(ψ̃) − (divgϕ̃ X + 1)ψ̃,
where we use that θX + X(ϕ̃) is a potential for X with respect to gϕ̃ .
1,α
Lemma 5.4.42. L̃ϕ̃ : X̃ 3,α → CX
(M ) is well-defined.
Proof. First note that as ψ̃ ∈ C 3,α (M ) and ϕ̃ ∈ U 3,α clearly ∆gϕ̃ ψ̃ ∈ C 1,α . Secondly X(ψ̃) ∈
C (M ) since ψ ∈ X̃ 3,α . And divgϕ̃ X = ∆gϕ̃ (θX + X(ϕ̃)). As θX is smooth and all second or higher
derivative of θX are uniformly bounded ∆gϕ̃ θX ∈ C 1,α (M ). By the estimates in the proofs of theorems
5.4.1 and 5.4.2 ϕ̃ ∈ U k,α for all k > 0 and in particular ∆gϕ̃ X(ϕ̃) ∈ C 1,α (M ). It is finally easy to check
L=(X) L̃ψ̃ = 0 and so the lemma follows.
1,α
Lemma 5.4.43. The following function spaces are equal and their norms are equivalent:
k,α
k,α
exp(−θX − X(ϕ̃))C k,α (M ) = exp(−X(ϕ̃))Cw
(M ) = Cw
(M )
and
exp(−θX − X(ϕ̃))X̃ k,α = exp(−X(ϕ̃))X k,α = X k,α
Proof. By the previous estimates X(ϕ̃) ∈ exp(−µθX )C l,α (M ) for any µ < 1, l ∈ N, and g is
uniformly equivalent to gϕ̃ . So for u = exp(−X(ϕ̃))v, v ∈ X k,α we have
kukX k,α = k exp(−X(ϕ̃))vkX k,α ≤ Ck exp(−X(ϕ̃))kX̃ k,α kvkX k,α ≤ C 0 kvkX k,α
for some uniform constant C 0 . This proves that exp(−θX − X(ϕ̃))X̃ k,α ⊂ X k,α . The opposite inclusion
works similarly.
Corollary 5.4.44. L is an isomorphism if and only if L̃ is an isomorphism
Proposition 5.4.45. Let (M, J, g) and θX be as in theorem 5.4.4 and Ωκ := {x ∈ M : θX > κ}.
Suppose that g̃ is a Riemannian metric on M with kg̃ − gkC 2,α (Ωκ ) = exp(−µκ) for some µ ∈ (0, 1).
Then the operator
Ã : X̃ 3,α
u
1,α
→ CX
(M ),
7→ ∆g̃ u − X(u) − (n + 1)u
is an isomorphism.
Proof. See appendix.
So let us show that L̃ is an isomorphism. Clearly L̃ is injective since L is injective: Indeed, if L̃(ψ̃) = 0
then L(e−θX −X(ϕt0 ) ψ̃) = 0 and this implies that ψ̃ = 0 by the maximum principle. In order to show
that L̃ is an isomorphism we write
L̃ψ̃ = ∆t0 ψ̃ − X(ψ̃) − (n + 1)ψ̃ + (n − ∆t0 (θX + X(ϕt0 )))ψ̃ .
|
{z
} |
{z
}
=:Ã(ψ̃)
=:K̃(ψ̃)
Now Ã : X̃ → Z̃ is an isomorphism (and in particular Fredholm) by proposition 5.4.45 and so if we can
show that K̃ : X̃ 3,α → Ỹ 1,α is compact then L̃ is also an isomorphism (an injective Fredholm operator
with index 0). This follows from the fact that the index of an operator does not change under compact
deformations.
Lemma 5.4.46. The operator K̃ : X̃ 3,α → C 2,α (M ) → C 1,α (M ) is compact.
Proof. Let un be a sequence in X̃ 3,α with kun kX̃ 3,α = 1. Let K̃(un ) = γun where kγkC 2,α ({θX >κ}) ≤
0
1,α
C exp(−θX /2) . We can find a subsequence unj ∈ X̃ 3,α which converges in Cloc
(M ) (α0 < α) to
u ∈ C 1,α (M ) and kunj kC 1,α ≤ 1. We can choose an exhaustion Ωj of M and without loss of generality
we can assume that
1
kunj − ukC 1,α0 (Ωj ) ≤ .
j
5.5. OLD EXAMPLES OF EXPANDING KÄHLER RICCI SOLITONS
69
Moreover we may also assume that kγkC 1,α (M \Ωj ) < 1/j and so
kγ(unj − u)kC 1,α (M ) ≤ kγkC 1,α0 (M \Ωj ) (kunj kC 1,α0 (M ) + kukC 1,α0 (M ) )
+ kunj − ukC 1,α0 (Ωj ) kγkC 1,α0 (Ωj )
<
C
j
which proves the lemma.
Now we can apply the implicit function theorem to finish the proof of theorem 5.4.4.
Proof of theorem 5.4.5. assume that there exist two solutions ϕ1 , ϕ2 ∈ U. Set ψ := ϕ2 − ϕ1 ,
then ψ satisfies
√
¯ n
(ωϕ1 + −1∂ ∂ψ)
log
+ X(ψ) − ψ = 0,
(5.8)
ωϕn1
and so by the maximum principle it follows that ψ = 0, i.e. ϕ2 = ϕ1 .
5.5. Old Examples of Expanding Kähler Ricci Solitons
Examples of Kähler Ricci solitons were constructed in [Cao85], [FIK03], and [DW11]. All of the
examples reduce the soliton equation to an (or a system of) ODE. Cao constructed expanding Kähler
Ricci solitons on Cn with positive bisectional curvature. Theses examples were later generalized in
[FIK03]. These generalizations also include examples of nontrivial asymptotically flat expanding Ricci
solitons. The examples in the first two references were then further generalized in [DW11] to include
examples on complex line and vector bundles over products of arbitrary Kähler Einstein manifolds. Here
we only consider asymptotically Ricci flat expanders. So in the following we will try to identify those.
Example 5.5.1. On Cn there exists a ‘trivial’ expander: namely the standard euclidean metric along with
the potential |x|2 /2. Clearly this expander falls into our class of asymptotically Ricci flat expanders. This
is the only expanding Ricci soliton metric (up to isometries) on Cn which is asymptotic to the euclidean
metric.
The next examples are due to [FIK03]. It also follows from the main theorem in the previous chapter:
Example 5.5.2. Consider the cyclic group Γ ⊂ U (n) generated by the action on Cn which sends zl to
exp(2πi/k)zl , l = 1...n, where k ∈ N. Clearly this group acts freely on Cn \ {0} and so the quotient
Cn /Γ along with the euclidean metric is a smooth cone with the origin as its vertex. This singularity
can be resolved by one blow up and one obtains smooth manifolds, the total spaces of line bundles
L−k → CP n−1 . It follows that L−k admits an expanding Ricci soliton metric if and only if k > n.
For each such k there exists a rotationally symmetric expanding Kähler Ricci soliton metric g such that
(M, g) is asymptotic to (Cn /Γ, δ). Note that this example follows from the theorem above. One can
generalize this to line bundles over manifolds over Kähler Einstein manifolds Z with positive scalar
curvature as done in [DW11]. However, here one can see that the assumption that the contraction
of the zero section gives a Ricci flat cone is quite restrictive. These examples are then of the form
m/I(Z)
M = KZ
, where I(Z) is the Fano index of Z and m > I(Z). Note that the asymptotic cones of
these manifolds are not flat anymore. So they provide the first examples of Kähler Ricci solitons which
are asymptotically Ricci flat but not ALE.
Remark 5.5.3. Recall that if C(S) is a cone over a regular Sasaki manifold S then C(S) is automatically
a line bundle L over a compact Kähler manifold Z with c1 (L) < 0 (As −dη restricted to Z is a
curvature form of L). And it follows from the remarks in the beginning of this chapter that L admits an
expanding Kähler Ricci soliton metric if and only if c1 (L) = λc1 (KZ ), λ > 0 i.e. Z is a Fano manifold.
The first Chern class of M restricted to Z, of the total space of L is then c1 (M )|Z = −c1 (KM )|Z =
−c1 (L) + c1 (KZ ) = (−λ + 1)c1 (KZ ). And as we have seen before it follows that λ > 1.
The next example will show that a further generalization for vector bundles fails if we require that the
asymptotic cone is Ricci flat and the metric is constructed as in [DW11].
70
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
Example 5.5.4. This in fact a ‘non-example’: suppose Σ is a U (1)-bundle over Z := CP n1 × N n2 ,
where n1 > 0 and N n2 is a Fano manifold that admits a Kähler Einstein metric. Then we can find a
regular Sasaki-Einstein metric on the total space S of Σ → Z if Z is an Einstein metric with Einstein
constant 2(n1 + n2 ) and
c1 (Z)
c1 (Σ) = q
, q ∈ Z<0 ,
I(Z)
where I(Z) is the Fano index of Z. Since c1 (Z) = π1∗ c1 (CP n1 ) + π2∗ c1 (N ) we have I(Z) = gcd(n1 +
1, I(N )). Now suppose in addition that
p
−1 ∗
π1 c1 (CP n1 ) +
π ∗ c1 (N )
c1 (Σ) =
n1 + 1
I(M ) 2
then
q
1
p
q
=−
and
=
I(Z)
n1 + 1
I(N )
I(Z)
and so
I(N )
I(Z)
I(N )
q=−
=p
, i.e. p = −
.
n1 + 1
I(N )
n1 + 1
q/I(Z)
We would need for KZ
to admit an expanding Kähler Ricci soliton metric q/I(Z) < −1 i.e.
p/I(N ) < −1 and hence −1/(n1 + 1) < −1 which is a contradiction.
The assumption on c1 (Σ) is motivated as follows: note that the line bundle associated to Σ is not a
minimal resolution. In fact, a CP n1 can be blown down and what remains is a rank n1 − 1 vector bundle
N . However if we allow k > 1
k
p
c1 (Σ) = −
π1∗ c1 (CP n1 ) +
π ∗ c1 (Z)
(n1 + 1)
I(N ) 2
then
kI(Z)
I(Z)
q=−
=p
n1 + 1
I(N )
which implies
kI(N )
.
p=−
n1 + 1
But then it follows from q/I(Z) < −1 that k > I(Z). So the line bundle associated to Σ is indeed a
minimal resolution.
Here is an example that does not require that the zero section admits a Kähler Einstein metric:
Example 5.5.5. Consider Z := CP 2 ]CP 2 and the total space KZp , p > 1 (p ∈ I(Z)−1 Z+ ). Then the
blow down of this space along the zero section is a complex cone C that admits a Ricci flat cone metric
(see [FOW09] Corollary 1.3). Note, however, that the Sasaki Einstein metric on S cannot be of the
form π ∗ h + η ⊗ η (i.e. regular) where η is some U (1)-connection on the U (1)-bundle associated to KZp .
Nevertheless, C \ {o} is biholomorphic to KZp \ {zero-section}. In other words KZp is a resolution of
C. It can be shown that the total space KZ admits a Ricci flat Kähler metric (since Z is toric, see
[Fut07]). The total space KZp on the other hand admits an expanding Kähler Ricci soliton metric for
p ∈ I(Z)−1 Z>I(Z) . The cone associated to M := KZp is the cone over S/Γ where S is the total space a
U (1)-bundle over Z and Γ ⊂ U (1) acts on S fiber-wise.
5.6. New Examples of Expanding Kähler Ricci Solitons
5.6.1. Toric Kähler Cones. To construct examples which are not rotationally symmetric (or
cohomogeneous one) we consider cones C(S) which are Ricci flat, Kähler and toric. A special class of
such cones are quotients of Cn by a finite abelian subgroup Γ of U (n) which acts freely on Cn \ {0}
equipped with the euclidean metric. Here we will first study toric Kähler cones and their resolutions in
general and then construct more concrete examples of expanders on toric resolutions of cones of type
Cn /Γ. For background on toric manifolds please refer to the appendix. Much of the following is inspired
by the work of van Coevering [vC10], [vC12].
5.6. NEW EXAMPLES OF EXPANDING KÄHLER RICCI SOLITONS
71
The idea to construct these example can be summarized as follows: We start out with a Ricci flat
Kähler cone. This cone is, together with the vertex, a complex variety and we look at the subclass
of toric varieties which are also toric. As such they posses a toric resolution which is a smooth (nonsingular) toric variety M . Then, finally, we construct a toric Kähler metric on M such that it satisfies
the assumptions of the theorem.
Definition 5.6.1. A Sasaki manifold (S, Φ, η, ξ, g) is called toric if the associated Kähler cone (C(S), J, ω, g)
is toric Kähler manifold, i.e. there exists an effective action of an n-dimensional torus Tn on C preserving the Sasaki structure such that the Reeb vector field ξ is part of the associated Lie algebra t of
Tn .
Given a toric Sasaki manifold we can associate a moment map µ to the action of Tn as follows. Let
ζ ∈ t, the Lie algebra of Tn , and Xζ its associated infinitesimal action on C(S). Let
µ : C(S) → t∗ ,
where
x 7→ (ζ 7→ hµ(x), ζi)
1
hµ, ζi := − r2 η(Xζ ).
2
Then
1
ıX d(r2 η)
2 ζ
1
1
= LXζ (r2 η) − dıXζ r2 η
2
2
1
1
= r2 LXζ η + dhµ, ζi
2
2
1
= dhµ, ζi.
2
n
So the action of T on C(S) is indeed Hamiltonian and µ is its associated moment map. The moment
cone is defined as the set
C(µ) := µ(C(S)) ∪ {0} ⊂ t∗ ≡ Rn .
ıXζ ω =
It can be shown that C(µ) is a strictly convex polyhedral cone, i.e. C(µ) is of the form
C(µ) :=
d
\
{y ∈ t∗ : huj , yi ≥ 0}
j=1
for some uj ∈ ZTn := Ker(exp(2π·) : t → Tn ), j = 1, ..., d, such that the uj ’s are primitive, i.e. not
integer multiples (p 6= ±1) of integer vectors, and the set {uj } is minimal in the sense that removing one
of the uj ’s changes the set C(µ). The cone is said to be non-singular if each sub-collection uj1 , ..., ujk
is art of a Z-basis of ZT , i.e. the real subspace spanned by uj1 , ..., ujk intersected with ZT is just the
Z-lattice generated by uj1 , ..., ujk . Let ξ be the Reeb vector field of S. It can be identified with an
element of t. Then µ(S) = {y ∈ C(µ) : y(ξ) = 12 } and S is smooth if and only if C(µ) is non-singular.
Let Int(C(µ)) be the open interior of C(µ). Then µ−1 (C(µ)) is an open and dense subset of C(S) on
which the torus act freely and biholomorphically. The dual cone is the given by
C(µ)∗ := {x ∈ t : hx, yi ∀y ∈ C(µ)}.
The Kähler cone C(S)∪{o} can also be described as a (singular) toric variety. Namely, u1 , ..., ud generate
a fan F whose highest dimensional cone is C(µ)∗ . And as shown in the appendix this fan gives rise to a
toric variety XF which is isomorphic to C(S) ∪ {o}.
5.6.2. Resolutions of toric Kähler Cones. As a singular toric variety a toric Kähler cone admits
a toric resolution. More precisely, the fan F associated with C(S) has a subdivision F 0 such that F 0 is
non-singular (see appendix C for details). Then the toric variety XF 0 associated F 0 is a resolution of
XF = C(S) ∪ {o} and the morphism π : XF 0 → XF is constructed from the inclusion ı : F → F 0 . The
non-singular condition on F 0 implies that M := XF 0 is non-singular i.e a smooth manifold. Moreover,
72
5. EXISTENCE OF EXPANDING KÄHLER RICCI SOLITONS
the action of the algebraic torus (C∗ )n on C(S)
√ lifts to M , in particular there exists a holomorphic
vector field X on M such that π∗ X = (r∂r − −1J∂r )/2 on the smooth part of C(S).
Example 5.6.2. If n = 2 any toric Ricci flat Kähler cone is of the form C2 /Γ where Γ is some finite
commutative subgroup of U (2) acting freely on C2 \ {0} and Kähler structure is given by the standard
Kähler structure on C2 . Here we study the case when Γ is of the form
* 2πij
!
+
e q
0
Γ=
| j = 0, ..., q − 1
2πijp
0
e q
where gcD(p, q) = 1. The associated moment map (with an appropriate choice of basis for t) is then
p
1
µ(z1 , z2 ) =
|z2 |2 + |z1 |2 , |z2 |2 .
q
q
And so
C(µ) = {σ1 (pe1 + qe2 ) + σ2 e1 : σ1 , σ2 ∈ R≥0 }
and
C(µ)∗ = {σ1 (qe∗1 − pe∗2 ) + σ2 e∗2 : σ1 , σ2 ∈ R≥0 }
Since gcD(p, q) = 1 it follows that the fan is generated by the integer vectors (q, −p) and (0, 1). Clearly
this fan is singular. Now there exists a general procedure to refine this fan to become non-singular. We
will look at the case p = 1. Then adding the edge generated by (1, 0) gives a fan F 0 which is indeed
non-singular. The associated toric variety XF 0 is then a smooth toric resolution of C2 /Γ, indeed it is
L−q → CP 1 . If q > p > 1 then the necessary refinement requires more steps. It is determined by the
continued fraction expression p/q. In particular if p = q − 1 then the refinement of the fan gives rise to
the resolutions in Kronheimer’s construction of Ricci ALE Kähler metrics in [Kro89a].
We will try to determine whether the resolution of the toric Kähler cone satisfies the condition of the
existence theorem for expanding Kähler Ricci solitons. As the toric action on the singular variety C(S)
is lifted to the resolution and since the radial holomorphic vector field is contained in the complex torus
which acts holomorphically on C(S) it clearly has an extension on any toric resolution M . And any
toric Kähler metric on M will make the lift a gradient vector field via its moment map. So it remains
to find a condition under which the resolution admits a toric Kähler metric ω such that [ω] ∈ Hc1,1 (M )
and there exists a smooth real-valued function f on M such that
√
¯
ω + ρ(ω) = −1∂ ∂f.
Recall that the fan gives back a complex toric variety. And any associated Delzant polyhedron (or
its generalization for the non-compact case) gives a toric Kähler structure on that variety. This is
described in more detail in the appendix. So suppose we have a refined fan F 0 . Then an associated
Delzant polyhedron is of the form
∗
P := ∩N
r=1 {y ∈ t : lr (y) ≥ λr },
where lj (y) = huj , yi, uj ∈ Zn and u1 , ..., uN generate F 0 . And in order for the associated Kähler form
ωP to satisfy (5.6.2) we need λk = 2π + hc, uj i for all j ∈ {1, ..., N } and some vector c (see appendix C
section 3.1). Whether P is still a Delzant polytope remains to be checked. This can be done rather easily
in case of example 5.6.2. As explained in the appendix the fan F 0 of the Hirzebruch Jung resolution of
C2 /Γ(p, q) is generated by e1 and pj e1 − qj e2 where p0 = p, q0 = q. And (pj , qj ) are generated according
to the recursion formula above (C.2.3). It is now a combinatorial problem (see appendix C) to prove
that P is a Delzant polytope if and only if rk > 2 for all k = 1, ..., j where rk are the integers showing
up in the continued fraction expression
1
p
= r1 −
.
q
r2 − ...−1 1
rj
These numbers correspond to the diagonal elements of the intersection matrix. In summary we have
5.6. NEW EXAMPLES OF EXPANDING KÄHLER RICCI SOLITONS
73
Theorem 5.6.3. Let p > q such that gcd(p, q) = 1 and rk > 2 for all k = 1, .., j. Then the minimal
resolution of C2 /Γ(p, q) admits an expanding Kähler Ricci soliton metric.
CHAPTER 6
Open Problems
In this chapter we would like to discuss some further directions for research, open questions and partial
results that we have not presented so far.
(1) Given any Ricci flat Kähler cone. When does it have a resolution as described in the previous
chapter? Which orbifold C2 /Γ, Γ ⊂ U (2) (or more generally R4 /Γ, Γ ⊂ SO(4)) admits an
expanding Kähler Ricci soliton coming out of it?
We have seen various examples of resolutions of Ricci flat Kähler cones with KM > 0. We can
also exclude several cases. In complex dimension 2 a minimal resolution is unique. Moreover,
all Ricci flat cones are of the form C2 /Γ, Γ ⊂ U (2) and |Γ| < ∞. For Γ ⊂ SU (2) the resolutions
are well known. These resolutions admit Ricci flat ALE Kähler metric and in particular none
of them (except C2 itself) admit an expanding Kähler Ricci soliton metric as KM is trivial.
When Γ is cyclic we already have necessary and sufficient conditions for the existence of an
expanding Kähler Ricci soliton metric. It remains to investigate the cases when Γ is not cyclic
and not a subgroup of SU (2). Finally there might exist expanders on deformations of C2 /Γ.
These manifolds are not resolutions of of the cone and in particular not biholomorphic to the
cone outside a compact set.
(2) Is a Kähler Ricci flow coming out of a given Käher cone an expanding Kähler Ricci soliton?
Are general Ricci flows coming out of (non-Ricci-flat) cones asymptotically self-similar?
(3) Define a relative expander entropy (see [FIN05]) for Ricci flows coming out of cones.
Recall that the expander Ricci functional W+ constructed in [FIN05] is infinite on noncompact expanding Ricci solitons. We have shown, however, that a Ricci flow coming out of
a Ricci flat cone approaches the original cone at infinity at exponential rate. So in principle it
seems likely that one should be able to define a relative expander entropy ν+ that compares
the evolving manifold with the original cone. This has been done for the mean curvature flow
[Ilm]. In [Ilm95] the existence of an expander for non-minimizing cones is shown by constructing competitors which minimize the expander functional for mean curvature flow. We
believe that a similar procedure should work for the Ricci flow as well. The stability of a cone
rel
should be defined in terms of the second variation of ν+
or its maximality respectively. It
should also help to show that each flow coming out of a cone converges to an expanding Ricci
soliton.
(4) Prove the existence of expanding Ricci solitons coming out of non-Ricci-flat Kähler cones. In
particular on cones over Sasaki Einstein manifolds.
The construction of expanding Ricci solitons in chapter 5 should be extended to the case
when the original cone is not Ricci flat. In this case we do not expect that [ω] ∈ Hc1,1 (M ).
75
76
6. OPEN PROBLEMS
Moreover, the expander metric g approaches the cone metric gC only at quadratic rate. Examples of such expanding Kähler Ricci solitons for general Kähler can be found in [FIK03]
and [DW11]. We plan to investigate the existence of non-singular expanding Kähler Ricci
solitons for general Kähler cones in the future. Futaki and Wang constructed singular expanding Kähler Ricci solitons (or rather incomplete flows with boundary) coming out of cones over
Sasaki manifolds which admit a transversal deformation to a Sasaki Einstein structure but
are not Einstein themselves. This is also related to the program of Song and Tian: there,
expanding Ricci solitons on small resolutions are needed to model the behavior of the Kähler
Ricci flow after singularities which are not log-terminal. These cones are not Ricci flat and the
first Chern class of the resolution is not an element of Hc1,1 (M ) in general.
(5) Are the expanding Kähler Ricci solitons constructed in the previous chapter unique?
We have shown that on a fixed resolution M of a given Ricci flat Kähler cone and a fixed
holomorphic vector field the expanding Käher Ricci soliton metric is unique. We expect that
any two expanders with different holomorphic vector fields on the same resolution are related
by a biholomorphism. This is true for shrinking Kähler Ricci solitons on compact Fano manifolds [TZ00]. For a given cone it should also be possible that expanding Ricci solitons live on
deformations and not a resolution.
(6) Formulation of an existence theory for (Kähler) Ricci flows on spaces with conical singularities.
There is a theory of Ricci flow on orbifolds as well as Ricci flows on certain singular algebraic varieties. Motivated by the program of Tian and Song it seems desirable to develop a
general theory of Kähler Ricci flows that smooth out certain singularities while allowing others
to persist.
(7) Do the expanding Ricci solitons constructed in the previous chapter violate the positive mass
theorem? In other words do there exist asymptotically conical metric on these manifold with
nonnegative scalar curvature and negative mass.
Le Brun [LeB88] showed that on L−k → CP 1 there exist complete ALE metrics with m(g) < 0
and vanishing scalar curvature. In [CS04] and [Joy95] examples of scalar flat ALE metrics
on resolutions of more general quotients of C2 are constructed. In both cases we show that
there exist expanding Kähler Ricci soliton metrics on these manifolds. This should also be
compared to Lagrangian mean curvature on manifolds with singularities which are modeled on
special Lagrangian cones. Special Lagrangian cones are area minimizing and therefore stable.
In other words, these singularities will not be smoothed out by mean curvature flow.
(8) Which Ricci flat (Kähler) cones are singularities of smooth (possibly non-Kähler) Ricci flows?
It is unlikely that a Ricci flat cone is a finite time limit of a smooth Ricci flow. Note that in
contrast the heat equation admits solutions which become stationary in finite time.
(9) Which Ricci flat (Kähler) cones have a forward solution of the Ricci flow?
Not every unstable Ricci flat cone can have a smooth forward evolution. In [HHS11] it is
6. OPEN PROBLEMS
77
pointed out that a Riemannian cone over a compact four dimensional Einstein manifold which
does not bound a 5-dimensional manifold cannot have a smooth forward evolution. As seen
before the link of a Kähler cone always bounds a a pseudoconvex domain in a smooth complex
manifold. However if KM ≤ 0 numerically we do not expect any Kähler Ricci flows coming out
of the cone which are not the cone itself. However it might be possible that either the Ricci
flow is not Kähler, at least not with the complex structure from the resolution.
APPENDIX A
Schauder Estimates for Ornstein Uhlenbeck Operators
In this appendix we derive the global Schauder estimates which are essential in chapter 5 from a similar
result of Lunardi [Lun98] for Hölder functions on Rn
A.1. Global Schauder estimates on Rn
In [Lun98] the following operator on Rn are considered:
Au(x) :=
X
k,l
Qkl (x)
X
∂u
∂2u
P k (x)
(x) +
(x) + r(x)u(x)
∂xk ∂xl
∂xk
k
such that there exists a uniform constant D > 0 with
• D−1 δij ≤ Qij ≤ Dδij as matrices, kQkC 3,α < D
• |P k (x)| ≤ D(1 + |x|), k∇P j kC 2,α (Rn ) < D,
• supx∈Rn r(x) = r0 < ∞, kr(x)kC 3,α < D.
Moreover, the following assumption is made:
For any λ > r0 there exists a smooth function ϕ : Rn → R such that limx→∞ ϕ(x) = ∞ and
supx∈Rn Aϕ(x) − λϕ(x) < ∞.
Under these assumptions the following theorem is proven
Theorem A.1.1 ([Lun98]). For any α ∈ (0, 1), λ > r0 , and f ∈ C 0,α (Rn ) there exists a unique solution
u ∈ C 2,α (Rn ) of Au − λu = F and a uniform constant C = C(n, α, λ, D) such that
kukC 2,α (Rn ) ≤ CkF kC 0,α (Rn ) .
A.2. Set up for M with a conical end
Let (M, g) be a smooth Riemannian manifold and (C, gC ) a Riemannian cone over (S, gS ) such that
there exists a compact subset C1 ⊂ M and a diffeomorphsim Ψ : M \ C1 → S × (1, ∞) such that
Ψ∗ g = gC . Let {Uj0 } be an open covering of S and {Uj , ϕj : Uj → Vj ⊂ S n−1 } an atlas on S such that
Uj0 b Uj and (ϕj )∗ gS is uniformly equivalent to gS n−1 . Let Ωj := Uj × (1, ∞) and Φj : Ωj → Vj × (1, ∞)
be defined by Φj (ξ, r) := (ϕj (ξ), r).
Lemma A.2.1. There exists a function θj (r, ξ) = ρ(r)χj (ξ) ∈ C 2,α (Ωj ) ∩ C ∞ (Ωj ) such that
(1) χj ∈ Cc∞ (Uj , [0, 1]), χj (ξ) = 1 for all ξ ∈ Uj0 ,
(2) ρ ∈ C ∞ ((1, ∞), [0, 1]) such that ρ(r) = 0 for r ∈ (1, 2] and ρ(r) = 1 for r ∈ [10, ∞),
(3) krρ0 (r)kC 0,α ≤ 2,
for some uniform constant C.
Proof. The existence of ρ is obvious and χj , too, as Uj0 is relatively compact in Uj .
79
80
A. SCHAUDER ESTIMATES FOR ORNSTEIN UHLENBECK OPERATORS
A.3. Global Schauder estimates on M
Let (M, g) and Φj be as above. Then (Φj )∗ g is uniformly equivalent to δ on Vj ×(1, ∞) and k(Φj )∗ gkC 4,α <
C. Let g̃ be a continuation of (Φj )∗ g to Rn such that g̃ is uniformly equivalent to the euclidean metric
and bounded in its derivatives. Consider the following operator on M :
Au := ∆u − X(u),
where X is a smooth vector field on M such that Ψ∗ X = 21 r∂r on M \ C1 . We prove
Theorem A.3.1. Let α ∈ (0, 1), µ > 0, and F ∈ C 0,α (M ). Then there exists a unique solution
u ∈ C 2,α (M ) to Au − µu = F and a uniform constant C = C(α, µ, n, g) such that
kukC 2,α (M ) ≤ CkF kC 0,α (M ) .
Proof. We prove the estimate first. So assume that u ∈ C 2,α (M ) is a solution. Let θj be as above
and let uj := θj u. The uj satisfies
∆uj − X(uj ) − µuj = F θj + Aθj + 2h∇θj , ∇ui =: F̃j .
Now let vj be the continuation of uj ◦ Φ−1
by zero to Rn . Then vj solves
j
X
X
∂ 2 vj
∂vj
kl
m ∂vj
Ãvj − µvj =
g̃
− Γ̃kl
−
xk
− µvj = F̃j .
∂xk ∂xl
∂xm
∂xk
k,l
k
It can be easily checked that Ã satisfies the assumption of theorem A.1.1 (see lemma below) and so
kvj kC 2,α (Rn ) ≤ CkF̃j kC 0,α (Rn ) .
So let us estimate kF̃j kC 0,α (Rn ) . Clearly
kF̃j kC 0,α (Rn ) ≤ C kF kC 0,α (M ) + kukC 1,α (Uj ×(1,10)) .
So it remains to show that kukC 1,α (Uj ×(1,10)) can be uniformly bounded by kF kC 0,α (M ) . Indeed, it
follows from the usual Schauder estimates, that
kukC 2,α (C1 ∪∪j Uj ×(1,10)) ≤ C kukC 0 (C1 ∪∪j Uj ×(1,20)) + kF kC 0,α (M ) .
And the estimate for kukC 0 follows from the maximum principle. Summing up over j we get
kukC 2,α (M ) ≤ CkF kC 0,α (M ) .
Uniqueness follows directly from the estimate. So let us now show existence of a solution u ∈ C 2,α (M ).
To this end, let σk (r) = 1 for r < k, σk (r) = (k + 1)/r for r > (k + 1) such that |σk0 | ≤ 2 and let
Xk := σk X. Then it can be shown by standard theory that there exists a solution uk ∈ C 2,α (M ) to the
equation
∆u − Xk (u) − µu = F.
By the estimates above there exists a uniform (independent of k) bound for kuk kC 2,α (M ) and so uk
2,β
converges in Cloc
(M ) to a function u ∈ C 2,α (M ) which is a solution of Au − µu = F .
Lemma A.3.2. Let Ã be as above. Then for any λ > 0 there exists a function ϕ such that limx→∞ ϕ(x) =
∞ while sup Ãϕ < ∞.
Proof. Take for example ϕ = |x|2 . Then clearly limx→∞ ϕ(x) = ∞ and
sup Ãϕ = sup −(1 + λ)|x|2 + O(|x|) < ∞.
Corollary A.3.3. Let M , X, and A as above and g asymptotically conical at exponential rate. Then
for all α ∈ (0, 1), µ > 0 and f ∈ C 0,α (M ) there exists a unique function u ∈ C 0,α (M ) such that
A − µu = f and a uniform constant C = C(α, µ, n, g) such that
kukC 2,α (M ) ≤ CkF kC 0,α (M ) .
A.3. GLOBAL SCHAUDER ESTIMATES ON M
81
Corollary A.3.4. Let (M, g), X, and A be as in the previous corollary. Then A : D(A) → C 1,α (M ),
where
D(A) = {u ∈ C 3,α (M ) : X(u) ∈ C 1,α (M )}
is an isomorphism and there exists a uniform constant C such that
kukC 3,α (M ) ≤ CkF kC 1,α (M ) .
APPENDIX B
Kähler cones as complex varieties
B.1. General Kähler cones
We briefly describe some concepts from algebraic geometry that we need to construct smooth varieties
asymptotic to cones. For more details we refer to [GH94] and [Dem12]. We will try to expand a little
bit more on anything that cannot be found there. All of this is standard knowledge for anyone familiar
with algebraic geometry. The application of these results to Kähler cones is due to van Coevering, most
importantly, the proof that any Kähler cone can be understood as a singular complex variety can be
found in [vC10]. As such any given Kähler cone can be desingularized (resolved) and the algebraic
methods involved in this procedure helps to determine whether certain obstruction for the existence of
a Kähler Ricci soliton metric on a resolution for a given Kähler cone vanish or not. There are different
notions of complex varieties. Here we will use the following definitions:
Definition B.1.1. We define affine algebraic (analytic) varieties, abstract algebraic varieties, complex
analytic sets, and complex analytic spaces as follows:
(1) An algebraic set in CN is the set of common zeroes of a finite number of algebraic functions,
i.e. polynomials on CN .
(2) An affine algebraic variety is an irreducible algebraic set, meaning that it cannot be written as
the union of two Zariski-open proper subsets.
(3) An algebraic variety is the set of common zeroes of a finite number of polynomials on CP N .
(4) An analytic variety (or analytic set) is a subset of CN which is locally the set of common zeroes
of a finite number of holomorphic functions.
(5) An abstract algebraic variety V is a collection of affine algebraic varieties patched together via
local isomorphisms.
(6) A complex analytic space V is a locally compact Hausdorff space together with a sheaf of continuous functions OV , such that there exists an open covering {Ui } of V and homeomorphisms
φj : Uj → Vj onto a analytic sets Vj ⊂ CNj such that φ∗j : OVj → OV |Uj is an isomorphism
(OVj being the sheaf of holomorphic functions on Vj ).
Remark B.1.2. An algebraic variety as defined above is always analytic and projective. An abstract
algebraic variety is not necessarily projective though certainly a complex analytic space. The famous
theorem of Chow states that any projective analytic variety is in fact algebraic (and any quasi-projective
analytic variety is a Zariski-open subset of an algebraic variety). And finally any complex manifold is
a complex analytic space and any Hodge manifold (by Kodaira) or Moishezon and Kähler manifold is
an algebraic variety. There are also a number of famous examples of complex manifolds which are not
algebraic. Namely, certain complex tori and the famous K3 surfaces. In case of an algebraic variety V
the structure sheaf of V is the sheaf OV of algebraic functions whereas on an complex analytic space V
the structure sheaf OV is the sheaf of holomorphic functions. In either case (V, OV ) is a ringed space.
In general such varieties (complex analytic spaces) are not smooth. Indeed, all kinds of singularities can
appear. However, there is an important condition that gives the variety some regularity.
Definition B.1.3 (normal variety). An algebraic variety (a complex analytic space) V is called normal
if for each p ∈ V the ring (OV )p is normal (see [Dem12]).
As it turns out the class of normal varieties (complex analytic spaces) is flexible enough for most
purposes. For example normality guarantees that the set of singular points Vsing (which is analytic) has
codimension at least 2 (see [Dem12]).
83
84
B. KÄHLER CONES AS COMPLEX VARIETIES
Definition B.1.4. A complex analytic space V is called holomorphically convex if for every compact
subset K the holomorphic convex hull
K̂ := {z ∈ V : |f (z)| ≤ sup |f (x)| for all f ∈ OV (V )}.
x∈K
is compact.
Definition B.1.5. A complex analytic space V is called 1-convex if it admits a continuous function
which is strictly plurisubharmonic outside a compact set.
Theorem B.1.6 ([CM85]). A complex analytic space V is holomorphically convex if it is 1-convex.
Definition B.1.7. A complex analytic space V is called Stein if it is holomorphically convex and any
compact analytic subset in V is finite.
Theorem B.1.8 (See [Dem12]). A Stein space can be holomorphically embedded into CN for N large
enough.
Finally we need the following Cartan-Remmert-reduction theorem:
Theorem B.1.9 ([Rem56], see also [Cam13]). Let V be a holomorphically convex complex analytic
space. Then there exists a Stein space W and a proper surjective holomorphic map f : V → W such
that
(1) f∗ OV = OW .
(2) f has connected fibres.
(3) The pair (f, W ) is unique up to a biholomorphism.
And
Theorem B.1.10 (Riemann Extension Theorem). Let V be a normal complex analyitc space and A
an analytic subset such that dim(A) ≤ dim(X) − 2. Then every holomorphic function on V \ A has a
meromorphic extension to V .
The following theorem appeared in [vC10].
Theorem B.1.11. Let (C(S), IC , ωC , gC ) be the Kähler cone over a Sasaki manifold (S, Φ, η, ξ, g). Then
C(S) ∪ {o} is a normal complex analytic space
Proof. As the details of this proof can be found in [vC10] we only give a sketch. Let CR(D, J)
be the automorphism group of S, i.e. diffeomorphisms σ : S → S such that σ∗ D = D and σ∗ ΦD = J
for the given CR structure (D, J) on S. One can show that for a given Sasaki structure (Φ, η, ξ, g)
compatible with (D, J) the Reeb vector field ξ generates a one parameter subgroup which is contained
in a maximal torus Tk ⊂ CR(D, J) and so the closure of this subgroup is compact in Tk . If the given
Sasaki structure is quasiregular then the one parameter subgroup itself is already compact in Tk and
hence 1-dimensional. So if tk is the Lie algebra of Tk and Zkt the corresponding lattice then ξ ∈ Zkt ⊗ Q
if the Sasaki structure is quasiregular. If the Sasaki structure is not quasiregular then we can pick
ζ ∈ Zkt ⊗ Q such that η(ζ) > 0 and a compatible Sasaki structure (Φ0 , η 0 , ζ, g 0 ). Note that there is a
biholomorphism ψ such that on (C, IC ) which maps one Sasaki structure to the other (in particular
ψ∗ ξ = ζ). Moreover the new cone is a cone over S with a quasiregular Sasaki structure. And so there
exists a C∗ action on S which acts freely on C(S) up to a finite number of points which have a finite
stabilizer. Hence the quotient C(S)/C∗ is a complex orbifold Z and C(S) is the total space W of a
complex line bundle L over Z minus the zero section. This line bundle is negative and we can find a
hermitian metric h on L such that r02 = h(x)|z|2 where r0 is the radial coordinate of the latter cone and
z is the fiber coordinate of the line bundle. One can see that W is swept out by strictly pseudconvex
domains Wc := {x ∈ W : r0 (x) < c} hence W is holomorphically convex and it follows that Z ⊂ Wc for
some sufficiently large c and so there exists a Remmert reduction: a complex space Y , a discrete subset
D of Y , and a proper holomorphic map π : W → Y which maps W \ Z biholomorphically onto Y \ D.
Since Z is connected it follows that D is connected in other words just one point {p} and so we can give
C(S) ∪ {o} the complex structure of Y which agrees with the original complex structure on C(S). B.1. GENERAL KÄHLER CONES
85
Remark B.1.12. With some more effort it can be shown that C(S) ∪ {o} is actually an affine algebraic
variety in CN (see [vC11] and [CH12]).
On a smooth variety V there is a well defined notion of a canonical line bundle KV . To extend this to
the singular case we need to distinguish two different types of divisors
Definition B.1.13. Let V be a complex analytic space of dimension n. A Weil divisor is a locally finite
Z-linear combination of (n − 1)-dimensional analytic irreducible set Dj :
X
D :=
λj Dj , λj ∈ Z.
(B.1)
We denote the set of Weil divisors in V by W Div(V ).
P
Definition B.1.14. A Weil divisor is called principal if it is the of the form D =
ordDj (f )Dj for
some meromorphic function where the sum is over all irreducible analytic subsets of V . A Weil divisor
D is called Cartier if it is locally principal.
Remark B.1.15. Recall that ordV (f ) for f meromorphic and V a analytic subset in an open subset U
of an complex analytic space is defined as follows: let p ∈ V and V = {x ∈ U : g(x) = 0}. Then ordV is
defined to be the largest integer a such that in the local ring (OV )p at p f = g a · h for some h ∈ (OV )p .
These two definitions have the following natural extension
Definition B.1.16. A Q-Weil divisor is a Q-linear combination of (n − 1)-dimensional irreducible
analytic subsets. And Q-Weil divisor D is called Q-Cartier if there exists an integer p such that pD is
Cartier.
On a non-singular complex analytic space the two types of divisors are actually identical. Moreover,
there is a one-to-one relation between holomorphic line bundles on V and Cartier divisors. In particular,
on a non-singular variety V the canonical line bundle KV has an associated Cartier divisor which we
denote by KV as well an associated invertible sheaf OV (KV ) (the sheaf of holomorphic sections of KV ):
Definition B.1.17. A sheaf F of OV modules over a complex analytic space V is called invertible if it
is locally isomorphic to the sheaf OV .
The three concepts of Cartier divisors, line bundles, and invertible sheaves are more or less interchangeable. On normal (possibly singular) varieties the canonical divisor can be defined as follows. Let Vreg
be the regular part V . On this part KVreg can be defined as usual. Since the complement Vsing has
codimension at least 2 the canonical divisor KV can be defined to be the closure of KVreg on V and
hence becomes a Weil divisor in V (note that any divisor is uniquely determined by its restriction to the
complement of a codimension 2 analytic subset). The canonical divisor does not necessarily have to be
Cartier. But it can be checked whether KV is Cartier by investigating whether the sheaf ı∗ OVreg (KVreg )
is invertible. The following example shows that not every Weil divisor is Cartier:
Example B.1.18. (see [Dem12]) Let V be the singular quadric {(x1 , x2 , x3 , x4 ) ∈ C4 : x1 x2 −x3 x4 = 0}.
Then the subset A := {(x1 , 0, x3 , 0)} ⊂ V is clearly a Weil divisor. However it cannot be defined by
holomorphic function near the origin. To see this let B := {(0, x2 , 0, x4 )} and note that A ∩ B = {0}.
However if A was a level set f −1 (0) near the origin then codimV (f −1 (0)∩B) ≤ 2 which is a contradiction.
Here we used the fact that for any f ∈ OV (U ) (U ⊂ V open), f 6= 0 the set f −1 (0) is either empty or
it has dimension dim(X) − 1 (see [Dem12]).
Recall that a bimeromorphism between two complex analytic spaces X and Y is a meromorphic morphism F : X → Y which has a meromorphic inverse. A meromorphic morphism is an equivalence class
of morphisms F : X − → Y defined on the complement of a nowhere dense analytic subset such that
the closure of the graph of F is an analytic subset of X × Y . (Two morphisms are equivalent if they
agree on the complement of a nowhere dense analytic subset).
Definition B.1.19. Let V be a complex analytic space with a non-empty singular set Vsing . A resolution
(W, π) of V is a non-singular complex analytic space W together with a bimeromorphic morphism
π : W → V such that π : W \ π −1 (Vsing ) → V \ Vsing =: Vreg is an isomorphism (biholomorphism) and
86
B. KÄHLER CONES AS COMPLEX VARIETIES
P
E := π −1 (Vsing ) is a simple normal crossing Weil divisor. That is, E = i Ei where each Ei is smooth
and Ei1 ∩ ... ∩ Eit is a transverse intersection. E is called the exceptional divisor.
Definition B.1.20. A singularity x ∈ V of a complex variety X is called rational if for any resolution
π : W → V all the higher direct images (Ri π∗ OW )x = 0 for all i > 0.
For example it is shown in [KM98] that any quotient singularity is in fact rational. A proof for the
following simple criterion for rationality of an isolated singularity can be found in [KM98].
Theorem B.1.21. Let V be a complex analytic space and let Ω be a holomorphic n-form nowhere
vanishing in a deleted neighborhood U of an isolated singular point x ∈ V . Then x is a rational
singularity if and only if for a sufficiently small neighborhood U 0 of x
Z
Ω ∧ Ω̄ < ∞.
(B.2)
U0
B.2. Ricci flat Kähler Cones
Proposition B.2.1. [vC10] Suppose C(S) is a Ricci flat Kähler Cone. Then C(S)∪{o} is Q-Gorenstein
and o is a rational singularity.
A normal complex analytic space V is Gorenstein if it is Cohen-Macaulay and the canonical Weil divisor
KV is Cartier. It is called Q-Gorenstein if KV is instead Q-Cartier. In particular, a quotient Cn /Γ
is Gorenstein if and only if |Γ| < ∞ and Γ ⊂ Sl(n, C). It is Q-Gorenstein for any finite subgroup
of Gl(n, C) acting freely on Cn \ {0}. The property of being C-M is implied by the fact that the
singularity is rational (by theorem 5.10 in [KM98]). Once this is shown it remains to prove that KV
(V := C(S) ∪ {o}) is Q-Cartier. To this end it suffices to show that O(pKV ) is invertible for some p ∈ N
and since the the singularity {o} has codimension at least 2 this follows (by the Riemann extension
theorem) if we can show that ı∗ O(pKC(S) ) is invertible. By assumption there exists a non-vanishing
p
section Ωp of KC(S)
or some p ∈ N and so ı∗ O(pKC(S) ) is indeed invertible (in fact trivial). The
singularity is rational for the following reason. Since S is Einstein with positive scalar curvature by
Meyer’s theorem it has a finite fundamental group π1 (S). So its universal cover S̃ is a compact Sasaki
Einstein manifold and Ṽ := C(S̃) ∪ {o} is a normal complex variety. The cone C(S̃) admits a nowhere
vanishing holomorphic n-form and it satisfies (B.2). So o ∈ Ṽ is a rational singularity and it follows
that o ∈ V is a rational singularity as well as it is the image of a finite unramified morphism g : Ṽ → V .
The following propositions are due to van Coevering
Proposition B.2.2. Let π : M → C(S) ∪ {o} be a resolution of a Ricci flat Kähler cone C(S). Then
H k (M, OM ) = 0 for all k > 0 and in particular H∂1¯ (M ) = 0.
As o ∈ V := C(S) ∪ {o} is a rational singularity (Rj π∗ OM )o for j > 0 and V is a Stein space it follows
that H i (V, Rj π∗ OM ) = 0 for i, j > 0. The Leray spectral sequence implies that H i (M, OM ) = 0 for
i > 0.
Proposition B.2.3. Suppose that π : M → C is a resolution of a Ricci flat Kähler cone C. Let
E :=√ π −1 (o) be the exceptional set and let ω1 be a complete Kähler metric on M such that ω1 =
¯ ∈ A1,1 (M, C) for some smooth function f ∈ C ∞ (M, R) and β ∈ A1,1 (M, C) a closed
β + −1∂ ∂f
(1, 1)-form with compact support. Then there exists a complete Kähler metric ω0 on M and compact
subset K1 ⊂ K2 ⊂ M such that ω0 = ω1 on K1 and ω0 = π ∗ ωC on M \ K2 .
This has already been proven in [vC10]. For convenience we include the proof here. Note that it suffice
to assume that [ω1 ] ∈ H 2 (M, R).
Proof. Set M̄ := {x ∈ M : r(π(x)) ≤ 1} and let Ei be the prime divisors
Pin E. H2n−2 (M ) is
generated by the fundamental classes of Ei . Suppose [ω1 ] is the Poincare dual to
ai Ei for ai ∈ R. By
assumption there exists a compactly supported closed (1, 1)-form α such that [α] = [ω1 ]. Let β ∈ A1 (M )
¯ 0,1 = 0.
be a smooth 1-form with dβ = ω1 − α. Then if β = β 0,1 + β 1,0 is the decomposition of β then ∂β
0,1
¯ = β
So there exists a smooth complex valued function γ with ∂γ
since by the previous lemma
B.2. RICCI FLAT KÄHLER CONES
87
√
¯
H∂1¯ (M ) = H 1 (M, OM ) = 0. And it follows that −1∂ ∂2=(γ)
= ω1 − α.
Let f = r2 be the Kähler potential of the cone metric on C. Choose 0 < a1 < a2 . And let ν : R+ → R
be a smooth
function with ν(x) = x for x > a22 , ν 0 (x), ν 00 (x) ≥ 0 for all x and ν(x) = c for x < a21 .
√
¯
Then −1∂ ∂ν ◦ f ≥ 0 extends to a form on M . Finally choose b1 and b2 with a22 < b1 < b2 . Let φ be a
non-negative function of r with φ(r) = 1 for r < b1 and φ(r) = 0 for r > b2 . Then define u := 2φ=(γ)
and so
√
√
¯ + λ −1∂ ∂(ν
¯ ◦ f) > 0
(B.3)
ω0 := α + −1∂ ∂u
satisfies the requirement of the lemma for λ sufficiently large.
APPENDIX C
Toric Manifolds and Toric resolutions
To construct examples of expanding Kähler Ricci solitons we constructed toric resolutions of a given
cone such that they admit a background metric necessary to apply the theorem about the existence of
expanding Kähler Ricci solitons. In this appendix we fill in the details from the literature. Most of
it is quite well known. However, the generalization of the theory of symplectic toric manifolds to the
non-compact case (which we will need here) seems to be more recent and requires a bit more care.
C.1. Symplectic Toric Manifolds
In this section we discuss the famous result of Delzant which shows that toric symplectic manifolds are
characterized by its moment polytope. Here we only need one aspect of Delzant’s theorem: any polytope
(polyhedral set) P which satisfies certain conditions gives rise to a symplectic toric manifold (MP , ωP ).
Moreover, this manifold MP carries a “canonical” complex structure JP compatible with ωP .
C.1.1. Toric action and moment maps.
Definition C.1.1. An action of a Lie group G on a manifold M is a group homomorphism
φ:
G → Diff(M )
g 7→ ψg ,
where Diff(M ) is the diffeomorphism group of M . It is called a symplectic action on a symplectic
manifold (M, ω) if
φ:
G → Symp(M, ω)
g 7→ ψg ,
where Symp(M, ω) is the group of symplectomorphisms, i.e. φ∗g ω = ω.
Now let (M, ω) be a symplectic manifold and G a Lie group with an action ψ : G → Diff(M ) and let g
its Lie algebra and g∗ the dual. Let exp : g → G, then for an element η ∈ g we can associate a vector
field Xζ on M defined by
d
φexp(tζ) (x).
Xζ (x) :=
dt |t=0
Definition C.1.2. The action ψ is a Hamiltonian action if there exists a map
µ : M → g∗ ,
such that
dhµ(x), ζi = ıXζ ω(x).
and
µ ◦ φg = Ad∗g ◦ µ,
The map µ is called a moment map.
for all g ∈ G.
Note that it follows that ψg is a symplectomorphism. A special case which we will consider here is the
case when G = Tn = (S 1 )n .
Definition C.1.3. A symplectic toric manifold is a (compact) connected manifold M of even dimension
2n with a symplectic structure ω and an effective Hamiltonian actions of the torus Tn = (S 1 )n .
89
90
C. TORIC MANIFOLDS AND TORIC RESOLUTIONS
We will denote by ZTn the integral lattice Ker(exp : g → Tn ) and by Z∗Tn its dual which is called the
weight lattice of Tn .
Example C.1.4. Here are some familiar examples of toric manifolds:
(1) (S 2 , ωst ) where T1 = S 1 acts on S 2 by moving points along the latitudes.
(2) (CP 1 , ωF S ) where S 1 acts on CP 1 by sending [z0 : z1 ] to [z0 , eit z1 ].
(3) Note that the torus itself is not a toric manifold.
(4) CP n−1 , n ≥ 2 is toric and also CP 2 #kCP 2 is toric for k = 1, 2, 3.
An important observation is the following
Theorem C.1.5 (Atiyah, Guillemin, Sternberg). Let (M, Tn , ω) be a compact connected symplectic toric
manifold and µ an associated moment map. Then
(1) the level sets of µ are connected
(2) the image of µ is convex
(3) the image of µ is the convex hull of the images of the fixed point in M of the action.
The image of µ is called moment polytope.
C.1.2. Delzant Polytopes and construction of (MP , ωP ).
Definition C.1.6. A Delzant (or unimodular) polytope P ⊂ Rn is a polytope (polyhedral set) which
satisfies
• there are n edges meeting at each vertex p ∈ P (simplicity)
• the edges at each vertex p ∈ P are of the form p + tvi , vi ∈ Zn , i = 1, ..., n (rationality)
• for each vertex p ∈ P the vectors v1 , ..., vn from a Z-basis of Zn (smoothness).
It is a famous theorem of Delzant that not only any moment polytope is a Delzant polytope but also that
any Delzant polytope is realized by a unique (up to symplectomorphisms) symplectic toric manifold.
Theorem C.1.7 (Delzant). There exists a bijective correspondence between the set of compact toric
symplectic manifolds {(M, Tn , ω)} and the set of Delzant polytopes.
Here we need a version that is suited for the non-compact case. As mentioned in [KL09] in the noncompact case the image of the moment map µ does not need to be convex and the map µ : M/Tn → t∗
does not need to be an embedding. In [KL09] the authors show that the right analogue of the unimodular
polytopes described above are manifolds W with corners along with a locally unimodular embedding
ψ : W → t∗ . They go on to show that given such a locally unimodular embedding ψ : W → t∗
there exists a bijection between the set of isomorphism classes of symplectic toric manifolds (M, Tn , ω)
which satisfy µ = ψ ◦ π, (π : M → M/Tn = W ) and H 2 (W, ZT × R). Here we will need a somewhat
simpler version of this very general theorem. Namely we want to show that for convex unimodular
polyhedral subsets W of t (i.e. ψ is just the inclusion and hence a global embedding) then the set of
isomorphism classes of toric symplectic manifolds associated W is non-empty and at least one of those
can be described as a symplectic quotient of C N , i.e. it also carries a “canonical” complex structure.
Proposition C.1.8. Let P ⊂ t∗ be a unimodular polytope of the form
∗
P := ∩N
i=1 {u ∈ t : hu, vi i ≥ λi , vi ∈ ZTn }
then there exists a toric symplectic manifold (MP , Tn , ωP ) with a moment map µP such that µP (M ) = P ,
µ̄P : M/Tn → t∗ is an embedding, and µ : M → t∗ is proper. Moreover, (MP , ωP ) is a symplectic
reduction of CN with the standard symplectic structure.
Proof. The are numerous detailed descriptions of the proof in the literature. This part of the
proof of Delzant’s theorem works in the non-compact case without any major modifications. We will
sketch it here. Let v1 , ..., vN ∈ ZTn be the integral vectors described above. By assumption v1 , .., vN
span Zn and they are primitive. Now define the linear map π by
∼ ZTn ⊗ R, ei 7→ vi ,
π : RN → Rd =
C.2. TORIC VARIETIES
91
where ei , i = 1, ..., N are standard basis vectors of RN . Then π is surjective. And we denote its kernel
by k. So we have an exact sequence
0 −→ k −→ RN −→ Rn −→ 0.
Now RN is the Lie algebra of TN so we get an induced map π : TN → Tn . Its Kernel K is a (N − n)dimensional subtorus of TN with Lie algebra k. In other words
1 −→ K −→ TN −→ Tn −→ 1
is an exact sequence. Finally the associated sequence
0 −→ t∗ −→ (RN )∗ −→ k∗ −→ 0
is also exact and we denote the map (RN )∗ → k∗ by ı∗ . Now consider the standard action of TN on CN .
This is a hamiltonian action and its moment map µCN is given by
1
µCN (z) = − |z1 |2 , ..., |zN |2 + const,
2
N ∼ ∗
where const is some constant vector in R = Z N ⊗ R. Choosing
T
const := (λ1 , ..., λN )
we claim that
(1) Z := ı∗ ◦ µ−1
(0) is a smooth submanifold of CN and
CN
(2) K acts freely on Z.
Once these two claims are established the manifold MP := Z/K is smooth and the Marsden Weinstein
theorem (see [MS98]) guarantees the existence of a Tn equivariant symplectic structure ωP on MP
which pulls back to restriction of standard symplectic structure of CN to Z.
C.1.3. The canonical complex structure JP on MP . The construction described above not
only gives us a symplectic structure on MP but also a compatible complex structure JP on MP . Together
they determine a toric Kähler structure on MP :
Proposition C.1.9 ([Abr03]). There exists a canonical compatible complex structure JP on (M, ωP )
such that JP is Tn invariant.
C.2. Toric Varieties
Instead of fixing a symplectic structure ω on M one could also fix a complex structure on M and require
that the action of the torus Tn is holomorphic, i.e. preserves J. Such a manifold (variety) is called a
toric variety. More precisely
Definition C.2.1. A toric variety is a (normal) irreducible complex variety X together with an action
of an algebraic torus (C∗ )n which has an open dense orbit.
Remark C.2.2. Irreducible means that V cannot be written as the union of two proper Zariski closed
subsets. In general it is not required that the toric variety is normal. However, once V is non-singular
it is automatically normal. Recall that a normal variety is defined as follows: for any point x ∈ V there
exists an affine neighborhood Ux such that for any f in the field of rational functions C(U ) such that
f (z)m +
m−1
X
gj (z)f (z)m−j = 0,
gj ∈ C[U ]
j=0
it follows that f (z) ∈ C[U ] where C[U ] is the coordinate ring of U , i.e. C[U ] = C[z1 , ..., zN ]/IU , IU being
the generating ideal of U . From a geometric point of view normality is a kind of regularity assumption.
For example if dimC (V ) = 1 and V normal then V is smooth. In general any singular set in a normal
variety has at least codimension 2.
92
C. TORIC MANIFOLDS AND TORIC RESOLUTIONS
C.2.1. Cones and Fans. General toric varieties can be classified by their combinatorial data.
These data are called fans. In the following we show how a complex variety determines a fan and how
a fan gives rise to a toric variety.
Definition C.2.3. A convex polyhedral cone is a subset σ ⊂ Rn of the form
σ = {a1 v1 + .... + ad vd : a1 , ..., ad ≥ 0},
n
where v1 , ..., vd ∈ R are called the generators of σ. It is called a rational cone if v1 , ..., vd can chosen
to be in Zn . A rational cone has a minimal set of generators in Zn (the edges of σ). And σ is called
regular if its set of minimal generators form a Z basis of N . Moreover σ is called simplicial if its minimal
generators are linearly independent over R. The dual cone σ ∗ ⊂ (Rn )∗ of σ is defined by
σ ∗ := {u ∈ (Rn )∗ : hu, vi ≥ 0 ∀v ∈ σ}.
A face of a cone σ is a set τ = Hu ∩ σ for some u ∈ σ ∗ where
Hu := {v ∈ Rn : hu, vi = 0}.
A cone σ is called strongly convex if {0} is a face of σ.
Definition C.2.4. A fan F is a collection of cones {σ1 , ..., σl } such that
(1) Every σ ∈ F is a rational strongly convex polyhedral cone,
(2) For every σ ∈ F, each face of σ is also in F,
(3) For all σ1 , σ2 ∈ F the intersection σ1 ∩ σ2 is a face of each.
The support |F| of F is ∪σ∈F σ. And F(r) denotes the set of all r-dimensional cones. A fan is called
complete if |F| = Rn , it is called simplicial if every cone σ ∈ F is simplicial, and it is called regular if
every cone σ ∈ F is regular.
C.2.2. Construction toric varieties from fans. In this subsection we explain how a fan F gives
rise to a complex variety XF . There are various ways to do this on of them is to construct XF as a
quotient of CN , where N is the number of integral generator v1 , ..., vN of one-dimension cones σ ∈ F(1)
and let
(t1 , ..., tN ) · (z1 , ..., zN ) 7→ (t1 z1 , ...., tN zN )
∗ N
be the standard action of (C ) on CN . As before let
π : ZN → Zn ,
ei 7→ vi .
Let k be the kernel of π and let K be the subgroup of (C∗ )N generated by the one-parameter groups
C∗ → (C∗ )n ,
τ 7→ (τ λ1 , ..., τ λN ),
with (λ1 , ..., λN ) ∈ k. Now for any I ⊂ {1, .., N } let
ZI := {z ∈ CN : zi = 0 ∀i ∈ I}
and
σI := {
X
vi : ai ≥ 0}.
i∈I
Set
U\
[
ZI .
σI ∈F
/
Then the subgroup K acts on U and we set formally
XF := U/K.
It is not clear that this is indeed a toric variety. However if F does not contain any toric factors (i.e. F(1)
spans Rn ) and F is simple then U/K is a geometric quotient, i.e. all K-orbits are closed. Moreover, XF
is a toric manifold with a toric action (C∗ )n ∼
= (C∗ )N /K. For details of this construction see [CLS11].
Theorem C.2.5. Given a fan F there exists a normal separated (i.e. Hausdorff in the classical topology)
toric variety XF . Conversely given a normal toric separated variety X there exists a fan F such that
X∼
= XF . Moreover,
C.2. TORIC VARIETIES
93
(1) XF is smooth (in particular a complex manifold) if and only if F is regular.
(2) XF is an orbifold if and only if F is simplicial.
(3) XF is compact if and only if F is complete.
C.2.3. Toric resolutions. Let Xsing (the singular locus) denote the set of all singular points in
an irreducible variety X and let Xreg := X \ Xsing be the smooth locus. Recall that
Definition C.2.6. Given an irreducible variety X, a resolution of X is a variety X 0 together with a
morphism π : X 0 → X such that
0
(1) X 0 is smooth (Xsing
= ∅) and irreducible.
(2) π is proper.
(3) π : X 0 \ π −1 (Xsing ) → X \ Xsing is an isomorphism.
It was proven by Hironaka that such a resolution exists for any (algebraic) variety over any algebraically
closed field with characteristic zero (and the prove was later extended to analytic varieties). The proof
is quite difficult although it has been subsequently simplified. For toric varieties, however, things are
much simpler and a resolution can be obtained by “improving” (refining) the fan F associated to a toric
variety X. A refinement of a fan F is a fan F 0 such that F ⊂ F 0 . The idea is that for any refinement
F 0 of F there exists a toric morphsim π : XF 0 → XF that makes (XF 0 , π) a resolution of XF . The
singular locus of XF can easily be determined by its fan. In fact,
[
Xsing =
V (σ).
σ is not smooth
Now the main theorem is the following
Theorem C.2.7 ([CLS11]). Every fan F has a refinement F 0 such that
(1) F 0 is regular.
(2) F 0 contains every cone of F.
(3) The toric morphism π : XF 0 → XF is a projective resolution.
The proof of this theorem can be found in [CLS11]. To find the refinement F 0 of F there exists a
general procedure which we will not describe here (see [CLS11] for more details). However, we will
discuss the the two-dimensional example of Hirzebruch Jung resolutions. We follow [Ful93] in this
description. Hirzebruch Jung strings or resolutions are resolutions of quotient singularities of the type
C2 /Zk where Zk is a cyclic subgroup acting freely on C2 \ {0}. Since Zk is a commutative subgroup of
U (2) the singularity is indeed toric and C2 /Zk is a (singular) affine toric variety with the associated fan
F = he2 , pe1 − qe2 i,
where e2 and e1 are the standard lattice vectors of ZT2 ⊂ t∗ and p and q are determined by the group
action of Zk which sends (z1 , z2 ) to (e2πi/p z1 , e2πiq/p ). Clearly the fan is not generated by a Z-basis of
ZT2 and hence XF is indeed singular. The first step to resolve this singularity is to refine F by inserting
another generator e1 . The cone generated by e1 and e2 is clearly non-singular. So if the refinement
F 0 is still singular so must be the cone generated by e1 and pe1 − qe2 . The new singularity is strictly
less singular (meaning the order of the group in the quotient is strictly less) than the first one. Indeed,
rotating the lattice by 90 degrees, i.e. sending e1 to e2 and e2 to −e1 and then applying the lattice
automorphism
1 0
A :=
−c 1
we can see that the new singularity correspond to the variety which belong to the fan (cone) generated
by e2 and p1 e1 − q1 e2 , where p1 = q and q1 = cq − p. Since q < p we can choose c to be the smallest
integer such that such that cq − p ≥ 0, call it r1 . It is easy to see that q1 = r1 q − p is strictly smaller than
q and so the singularity is of the form C2 /Z(q1 , p1 ). We can repeat this procedure inductively lowering
the group order until we end up with a smooth cone (which is the case when qj = 0). It follows that
94
C. TORIC MANIFOLDS AND TORIC RESOLUTIONS
the numbers ri can be determined by the continued fraction expansion of
p
1
.
= r1 −
q
r2 − ...−1 1
rj
The resulting resolution XF 0 associated to refined non-singular fan F 0 is simply connected. Moreover,
XF 0 contains r holomorphic spheres S1 , ..., Sr with the intersection matrix


−r1
1
0
···
0
 1
−r2
1
···
0 


 0
1
−r3 · · ·
0 
(Si · Sj ) = 
.
 ..
..
..
.. 
.
.
 .
.
.
.
. 
0
0
0
···
−rj
The exceptional set of the resolution is given by E = S1 ∪ ... ∪ Sr .
Remark C.2.8. Note that there is a slight ambiguity here. In fact, if the process described above is
applied straightforward it does not (sometimes) seem to give a refinement of the original fan. However,
it can be shown that the toric variety associated to the fan F̃ generated by {e2 , p̃e1 − q̃e2 } is isomorphic
to the XF if p̃ = p and q̃q ≡ 1(p). One can check that this implies that the resolution does not depend
on the order of r1 , ..., rj . Unfortunately this simple algorithm to produce a toric resolution does not
work in higher dimensions anymore.
C.3. Toric Kähler geometry
The two concepts of toric manifolds, toric varieties on the one hand and toric symplectic manifolds on
the other, come together in the theory of toric Kähler manifold as follows.
Definition C.3.1. Given a moment polytope P of the form
N
\
P :=
{u ∈ t∗ : hu, vi i ≥ λi },
i=1
where v1 , ..., vN ∈ ZTn and λi ∈ R. Then the faces of P consist of the sets
FI :=
N
\
{u ∈ t∗ : hu, vi i = λi , ∀i ∈ I},
i=1
where I ⊂ {1, ..., N }. The associated fan F(P ) is then
F(P ) := {σI : FI is a face of P },
where σI is the cone generated by vi , i ∈ I.
It follows that if P is of the form described above then (XP , ωP ) also has the structure of a toric variety
and hence a complex structure JF (P) determined by the associated fan F(P ). One can show that this
complex structure is indeed compatible with ωP and
Theorem C.3.2 ([Abr03]). The complex manifold (MP , JP ) is biholomorphic to the toric variety
XF (P ) .
To summarize: there exists a canonical Kähler structure on each toric manifold (MP , ωP ). In general
we define a toric Kähler manifold as follows
Definition C.3.3. A toric Kähler manifold is a connect 2n-dimensional Kähler manifold (M, J, ω)
equipped with an effective hamiltonian action
τ : Tn → Bih(M, J, ω)
of the standard torus Tn .
C.3. TORIC KÄHLER GEOMETRY
95
In particular, a toric Kähler manifold (M, J, ω, τ ) is a symplectic toric manifold. And as such it has
a moment polytope P (which is the image of the moment map if for example M is compact). Now it
follows from Delzant’s theorem that (MP , ωP , τP ) is symplectomorphic to (M, ω, τ ). Moreover, there
exists a canonical complex structure JP on MP which makes (MP , JP , ωP , τP ). As a complex manifold
(MP , JP , τP ) is biholomorphic to (M, J, τP ) however this map is usually not the symplectomorphism
which identifies ω and ωP , i.e. the two manifolds are not isometric as Kähler manifolds. In the following
we will describe how to obtain the canonical Kähler structure explicitly using action angle coordinates.
Let M := (M, J, ω, τ ) := (MP , J, ωP , τP ) be a toric Kähler manifold where (MP , ωP , τP ) obtained from
a polytope
N
[
P :=
{u ∈ t∗ : hu, vi i ≥ λi }
i=1
and J is a complex structure on MP compatible with ωP . Let µP its moment map. Let P o denote the
o
o
n
open interior of P and let M o = µ−1
P (P ). Then MP is open and dense in MP and T acts freely on
o
o
o
MP . We can choose (symplectic action-angle coordinates)coordinates (x, y) : M → P × Tn such that
ω=
n
X
dxi ∧ dy j .
j=1
In these coordinates the complex structure is of the form
0 −G−1
J=
G
0
and the integrability of J implies that Gij,k = Gik,j and so G = Hess(g) and the Riemannian metric is
G
0
ω(J· , ·) =
.
0 G−1
As mentioned above the Delzant construction induces a canonical complex structure and it was determined by Guillemin that there corresponding complex potential is
N
gP (x) =
1X
lj (x) log(lj (x)),
2 j=1
where lj (x) = hx, vj i − λj . Moreover any compatible J is determined by a potential of g the form
g = gP + h
where h is a smooth function on the closure of P o .
On the other hand we can choose complex coordinates (w, z) on M 0 ∼
= Cn /(2πiZ) = {w + iz : w ∈
R, z ∈ R/2πZ} and in these coordinates
0 −1
J=
1 0
and ω is given by its potential
√
¯
ω = 2 −1∂ ∂f.
Since ω is invariant under the action of Tn the function f depends only on the coordinates w1 , ...., wn ∈ R,
in other words f = f (w) ∈ C ∞ (Rn , R). In these coordinates
0
F
ω=
−F 0
where F = Hess(f ).
The two coordinate systems are related as follows: note first that the coordinate x is given by the
moment map x = µ(w, z). In complex coordinates the moment map is given by the relation
∂µi (w, z)
= ı ∂ ω = Fij
∂zj
∂wj
96
C. TORIC MANIFOLDS AND TORIC RESOLUTIONS
and so up to a constant
∂f
.
∂wi
In other words, (x, y) and (w, z) are related by a Legendre transformation and
x = µi (w, z) =
∂f
, y=z
∂w
maps Cn /2πiZ diffeomorphically onto P o × Rn /2πZ. Moreover,
X ∂f
∂g
f (w) + g(x) =
·
.
∂w
∂x
j
j
j
x=
It follows that the canonical Kähler potential is of the form
N
f (w) = −
N
1X
1X
lj (x) log(lj (x)) +
[(lj (x) + λj ) log(lj (x)) + (lj (x) + λj )]
2 j=1
2 j=1
N
=
1X
[λj log(lj (x)) + (lj (x) + λj )] .
2 j=1
Moreover, F and G are related by
Gij (x) = Fij (w),
where Gij = (G−1 )ij .
Proposition C.3.4 ([Abr03]). Let (MP , Jp , ωP , Tn ) be a toric Kähler manifold with the canonical
complex structure JP . Then

−1
N
Y
det(GP ) = δ(x)
lj (x) ,
j=1
where δ(x) is a strictly positive function which is smooth on the closure P .
In complex coordinates the Ricci form is given by
ρij = −2π
∂ 2 log(det(F ))
.
∂wi ∂wj
C.3.1. Geometric equations on toric manifolds. With the observations above several Kähler
geometric problems become non-linear scalar PDEs on polytopes in Rn . Also obstructions to the existence of solutions can be formulated in a combinatorial way. For example let (M, ωP , JP , Tn ) be a toric
Kähler manifold. Then the question whether there exists a toric symplectic form ω such that [ω] = [ωP ]
compatible with JP such that for some constant λ
ρ = λω.
Proposition C.3.5 ([Abr03]). Let (M, JP , ωP , Tn ) be a toric Kähler manifold.
Suppose that ϕ is a
√
¯ is still positive.
smooth function on M which is invariant under the torus action such that ω + −12∂ ∂ϕ
Then there exists a smooth function h : P → R such that the corresponding complex potential g is of the
form g = gP + h.
Now in complex coordinates the problem can be reformulated as
f = −πλ log(det(F )) + c(w),
where c(w) is an affine function. In symplectic coordinates this equation becomes
∂g
= πλ log(det(G)) + c(∇g).
∂xi
And using that g − gP =: h is smooth on P and
−g + xi
log(det(G)) = log(det(GP )) + log(δh ),
C.3. TORIC KÄHLER GEOMETRY
97
6
"
"
"
"
"
"
"
"
"
"
"
"
Figure 1. The polytope P associated to [ωP ] = −2πc1 (M ) on the resolution of C2 /Γ(11, 4).
where δh is a positive smooth function on P we get
∂gP
∂h
−gP + xi
− h + xi
= πλ log(det(GP )) + 2π log(δh ) + c(∇g).
∂xi
∂xi
And so we can conclude that a necessary condition is that
∂gP
h̃ := −gP + xi
− πλ log(det(GP )) − c(∇g)
∂xi
Q
is smooth on P . Now using that det(GP ) = j (lj (x))−1 δ(x)
X
1X
1X
λj log(lj (x)) + πλ
log(lj (x)) −
c(uj ) log(lj (x))
2 j
2
j
is smooth on P only if λj = −2πλ + c(uj ) for all j. In particular, [ω] = 2πλc1 (M ).
C.3.2. Kähler metrics on toric varieties. Not every toric variety XF can be equipped with a
toric Kähler structure. However if the fan arises from a polytope then this is possible. So given a fan F
generated by primitive elements uj ∈ F(1), i.e. uj ∈ ZTn , we have to figure out whether there exists a
good choice of λj such that the polyhedral set
P :=
N
\
{y ∈ t : huj , yi ≥ λj }
j=1
is a Delzant polytope. Once this is the case the Kähler metric on XF is given by the Delzant construction
and more concretely by Guillemin’s formula for the canonical symplectic potential.
Example C.3.6. Consider the example C2 /Γ(p, q) above with p = 11 and q = 4. It follows that r1 = 3
and r2 = 4. So the associated refinement F 0 is given by p1 = 4, q1 = 1 and p2 = 1, q2 = 0. So we can
construct our desired background metric on XF 0 as follows. Choose λ = −1, then we set
gP (x) := (y − λ0 ) log(y − λ0 ) + (11x − 4y − λ1 ) log(11x − 4y − λ1 )
+ (4x − y − λ2 ) log(4x − y − λ2 ) + (x − λ3 ) log(x − λ3 )
where λr = 2π, r = 0, .., 3 and P := {(x, y) ∈ R2 : lr (x, y) ≥ 0} (see figure 1).
APPENDIX D
Maximum Principles and Schauder Estimates
D.1. Maximum Principles
In this section we recall some well known maximum principles tailored for our purposes.
D.1.1. Elliptic Maximum Principles.
Theorem D.1.1 (Maximum principle on complex manifolds, [BCG+ 12]). Let (M, J) be a complex
manifold and f ∈ C 2 (M, R). Assume that f attains its maximum (minimum) at p. Then
√
¯ (p) ≤ 0 (≥ 0), and df (p) = 0.
−1∂ ∂f
Corollary D.1.2. Let (M, J, g) be a Kähler manifold and f ∈ C 2 (M, R). If f attains its maximum
(minimum) at p ∈ M then
∆f (p) ≥ 0 (∆f (p) ≤ 0).
Theorem D.1.3 (Elliptic Maximum principle on Riemannian manifolds, [BCG+ 12]). Let (M, g) be a
Riemannian manifold and f ∈ C 2 (M, R). Let Lu := ∆u + X(u). Assume that u attains its maximum
(minimum) at p then Lu(p) ≥ 0 (Lu(p) ≤ 0).
Corollary D.1.4. Suppose that (M, J, g) is a complete noncompact manifold, λ : M → R sucht that
inf λ = λ0 > 0, and limx→∞ u(x) = 0. If Lu ≥ 0 on M then sup u ≤ 0.
D.1.2. Parabolic Maximum Principle.
Theorem D.1.5 (Parabolic maximum principle, [BCG+ 12]). Let M be a smooth non-compact manifold,
ΩT ⊂ (0, T ] × M such that ΩT = M × (0, T ] \ K for some connected compact subset K ⊂ M × [0, T ],
{g(t) : t ∈ [0, T ]} a smooth family of uniformly equivalent complete Riemannian metrics on each time
slice of ΩT and u ∈ C 2 (ΩT ) ∩ C 0 (ΩT ) with max[0,T ] lim supx→∞ u(x, t) ≤ 0 satisfies
∂u
≤ ∆g(t) u
∂t
on ΩT . Let ∂ΩT := ∂K ∪ (Ω¯T ∩ M × {0}). Then
sup u ≤ sup u.
ΩT
∂ΩT
Proof. Let
− sup u(x, t).
T − t (x,t)∈∂ΩT
Assume that supΩT v > 0. Then the maximum has to be attained in the interior of ΩT in a point
(x0 , t0 ). And there ∂t v (x0 , t0 ) = 0 and ∆g(t0 ) v(x0 , t0 ) ≤ 0. At the same time
v (x, t) := u(x, t) −
∂v
≤ ∆g(t) v −
.
∂t
(T − t)2
And this implies that at (x0 , t0 )
∂v
(x0 , t0 ) ≤ ∆g(t0 ) v (x0 , t0 ) −
≤−
0=
∂t
(T − t0 )2
(T − t0 )2
which is clearly a contradiction. Hence
u(x, t) ≤
+ sup u(x, t),
T − t (x,t)∈∂ΩT
99
100
D. MAXIMUM PRINCIPLES AND SCHAUDER ESTIMATES
and since this is true for any > 0 the theorem follows.
D.2. Schauder Estimates
In this section we first state the famous local Schauder estimates that will be needed in chapter 5.
Theorem D.2.1 (Schauder-Morrey Estimates). Let B1 (0) and B2 (0) be two open balls in Rn and L be
a linear elliptic operator of second order on function on B2 (x) defined by
X
X
∂u
∂2u
Lu :=
a(x)ij
(x) +
b(x)j
(x) + c(x)u(x).
∂xi ∂xj
∂xj
i,j
j
Let α ∈ (0, 1) and assume that for some λ, Λ > 0
max{kaij kC 0,α , kbj kC 0,α , kckC 0,α } < Λ
and
inf
n
ξ∈R \{0}
X
aij ξi ξj > λ
for all x ∈ B2 then there exist constants C1 and C2 depending on n, α, Λ, λ such that if u ∈ C 2 (B2 ) and
Lu ∈ C 0,α (B2 ) then u ∈ C 2,α (B1 ) and
kukC 1,α (B1 ) ≤ C1 kLukC 0 (B2 ) + kukC 0 (B2 )
and
kukC 2,α (B1 ) ≤ C1 kLukC 0,α (B2 ) + kukC 0 (B2 ) .
For proofs see [Mor66] and [GT01].
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Acknowledgements
It is my distinct pleasure to thank several people. First of all I would like to express my deepest gratitude
to my advisor, Tom Ilmanen, for his enthusiasm, his guidance, the many hours of inspiring discussions
in his office, and sharing his seemingly endless supply of new ideas.
I would also like to thank my co-examiner Michael Struwe whose weekly “Gipfeltreffen” in the Dozentenfoyer and the coffee seminar made life at ETH so much more enjoyable. His support and interest
certainly helped me a lot.
I am very grateful to Lei Ni for spontaneously agreeing to be my second co-examiner and for his wonderful books on the Ricci flow which never left my desk throughout my studies.
I want to thank Robert Haslhofer for his collaboration on one paper and for giving several excellent
formal and informal talks at ETH on the Ricci flow and related topics. I would also like to thank him
for his patient explanations and his organisation of various informal seminars.
Thanks to Duong Phong, Valentino Tosatti and the members of the complex geometry group at Columbia
University for letting me speak at their complex PDE seminar.
A big thankyou goes to all members of group 6 for creating a very nice atmosphere. In particular I want
to thank my office mates, Jörg Hättenschweiler and Dominique Aberlin, for their interest and willingness
to engage in long discussions on mathematics and various other topics. I like to thank all past, current,
and visiting members of the coffee seminar, Melanie, Simon, Onur, Robert, Pierre, Michael, Jörg, Dominique, Luca, Franziska, Jonas, Federico, and Thomas, for giving several very interesting talks.
During my studies at ETH I have seen several group six organisers come and go. I would like to thank
Johannes Sauter, Remi Janner, Laura Keller, Cedric Membrez, and Micha Wasem for being excellent
“bosses”.
Throughout my years at ETH I had the opportunity to attend a number of outstanding lectures by
C. Bellettini, S. Brendle, Y. Brenier, M. Eichmair, J. Evans, P. Germain, A. Grigor’yan E. Hebey, T.
Ilmanen, O. Raphael, F. Pacard, T. Rivier, D. Salamon, W. Schlag, and M. Struwe which I am very
grateful for. I would like to thank Anna, Juan, and Mauricio for hosting me during my defense.
Very special thanks to my family for all their love and support.
And finally I would like to thank Aor for all the important things in life and Isara. You both mean
more to me than mathematics ever could.
107

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