Completions of regular ambitoric 4-manifolds

Completions of regular ambitoric
4-manifolds: Riemannian Kerr metrics and
beyond
Kael Dixon
Doctor of Philosophy
Mathematics and Statistics
McGill University
Montreal,Quebec
2016-07-12
A thesis submitted to McGill University in partial fulfillment of the requirements of
the degree of Doctor of Philosophy
c
Kael
Dixon 2016
ACKNOWLEDGEMENTS
I would like to thank my doctoral supervisors Niky Kamran and Vestislav Apostolov for suggesting this project for me, as well as providing guidance in both the
material and the style. I would also like to thank Yael Karshon for pointing me
in the direction of folded symplectic structures. Many thanks go out to my family,
whose love and support throughout the years has been invaluable. I would also like
to thank my friends and colleagues in Montreal who have helped foster my mathematical growth, especially Ben Smith and Omid Makhmali.
ii
ABSTRACT
We show that the conformal structure for the Riemannian analogues of Kerr
black-hole metrics can be given an ambitoric structure. We then discuss the properties of the moment maps. In particular, we observe that the moment map image
is not locally convex near the singularity corresponding to the ring singularity in
the interior of the black hole. We also study the Tomimatsu-Sato metrics, which
generalize the Kerr metrics. We show that these also admit Riemannian signature
analogues, and admit almost-complex analogues of ambitoric structures. We then
proceed to classify regular ambitoric 4-orbifolds with some completeness assumptions. The tools developed also allow us to prove a partial classification of compact
Riemannian 4-manifolds which admit a Killing 2-form.
iii
ABRÉGÉ
Nous montrons que la structure conforme pour les analogues riemanniens des
métriques trou noir de Kerr peut être munie d’une structure ambitorique. Nous
discutons ensuite des propriétés des applications moment. En particulier, nous observons que l’image de l’application moment n’est pas localement convexe près de la
singularité qui correspond à la singularité de l’anneau à l’intérieur du trou noir. Nous
étudions également les métriques de Tomimatsu-Sato, qui généralisent les métriques
de Kerr. Nous montrons qu’elles admettent aussi un analogue en signature reimannienne et un analogue presque complexe de la structure ambitorique. Nous classifions ensuite les 4-orbifolds ambitoriques réguliers avec quelques hypothèses de
complétude. Les outils développés nous permettent aussi d’obtenir une classification partielle des 4-variétés compactes riemanniennes qui admettent une 2-forme de
Killing.
iv
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
ABRÉGÉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
1.2
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3
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5
8
13
Moment maps and folded symplectic structures . . . . . . . . . . . . . .
18
2.1
2.2
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18
20
21
21
23
Classification of complete regular ambitoric 4-orbifolds . . . . . . . . . .
25
3.1
3.2
25
29
32
36
41
44
44
50
1.3
1.4
2
3
3.3
Introduction to toric geometry . . . . . . . . . . . . . . . . . . .
The Riemannian Goldberg-Sachs theorem and related geometric
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of results . . . . . . . . . . . . . . . . . . . . . . . . .
Ambitoric structures and ambitoric ansatz spaces . . . . . . . .
Folded symplectic structures
Normal forms . . . . . . . .
2.2.1 Parabolic type . . . .
2.2.2 Hyperbolic type . . .
2.2.3 Elliptic type . . . . .
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Gauge group for ambitoric ansatz spaces . . . . . . . .
Asymptotic analysis of ambitoric ansatz spaces . . . . .
3.2.1 Folds . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Edges . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Corners . . . . . . . . . . . . . . . . . . . . . . .
Classification of regular ambitoric orbifold completions
3.3.1 Local asymptotics . . . . . . . . . . . . . . . . .
3.3.2 Lokal-global-Prinzip . . . . . . . . . . . . . . . .
v
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3.3.3 Classification results . . . . . . . . . . . . . . . . . . . . . .
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Application: Compact 4-manifolds admitting Killing 2-forms . . . . . . .
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5
Tomimatsu-Sato metrics . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.1
5.2
5.3
5.4
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66
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Appendix: Busemann completions . . . . . . . . . . . . . . . . . . . . . .
89
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
5.5
6
Qualitative features in Lorentzian signature . . . . .
Wick rotation . . . . . . . . . . . . . . . . . . . . .
Factor structure . . . . . . . . . . . . . . . . . . . .
Symplectic forms . . . . . . . . . . . . . . . . . . .
5.4.1 Kerr Case . . . . . . . . . . . . . . . . . . .
5.4.2 Almost ambitoric structures . . . . . . . . .
5.4.3 Qualitative features in Riemannian signature
Moment maps . . . . . . . . . . . . . . . . . . . . .
5.5.1 δ = 2 case . . . . . . . . . . . . . . . . . . .
vi
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LIST OF FIGURES
Figure
page
2–1 The moment map near a fold. . . . . . . . . . . . . . . . . . . . . . .
20
2–2 The moment map on a parabolic ambitoric ansatz space. . . . . . . .
22
2–3 The moment map on a hyperbolic ambitoric ansatz space.
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23
2–4 The moment map on an elliptic ambitoric ansatz space. . . . . . . . .
24
4–1 The moment map image of CP2 with a hyperbolic ambitoric structure. 61
4–2 The moment map image of the kth Hirzebruch surface Hk with a
hyperbolic ambitoric structure. . . . . . . . . . . . . . . . . . . . .
62
5–1 Qualitative picture of Tomimatsu-Sato metrics (δ ≤ 3) in Lorentzian
signature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5–2 The domain of Riemannian Kerr. . . . . . . . . . . . . . . . . . . . .
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5–3 The momentum image of Riemannian Kerr. . . . . . . . . . . . . . .
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5–4 Qualitative picture of Tomimatsu-Sato metrics (δ ≤ 3) in Riemannian
signature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5–5 Moment map image of TS2. . . . . . . . . . . . . . . . . . . . . . . .
87
5–6 Moment map image near the fold of TS2. . . . . . . . . . . . . . . . .
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vii
CHAPTER 1
Introduction
The origins of this thesis lie in the similarity of classification results by both of
my supervisors and their collaborators. The setting is four dimensional geometry,
studying generalizations of Einstein metrics which have structures coming from generalizations of the Goldberg-Sachs theorem, but the signature of the metrics are not
the same. In [10], Einstein Lorentzian metrics with algebraically special Weyl curvature of Petrov type D are locally classified. If such a metric admits a Riemannian
invertible structure (an L ∈ SO(T M ) satisfying L2 = Id and L ◦ ∇LX L = ∇x L ◦ L),
then there exists a local 2-parameter isometry group G which is orthogonally transitive [11], meaning that each orbit is orthogonal to a family of surfaces. In Riemannian
signature, we will see in section 1.2 that Einstein metrics which have algebraically
special Weyl tensors of a certain type admit an ambikähler structure. A regular ambitoric structure is a type of ambikähler structure which admits a local 2-parameter
isometry group which is orthogonally transitive. In [3], Einstein Riemannian metrics
which admit regular ambitoric structures are locally classified. In fact, more general
classes of metrics were classified in both cases, but we focus on the Einstein case for
clarity. In both cases, despite very different techniques in the proof, both classifications result in metrics in the same form, depending on a handful of polynomial
functions, using coordinates (u, v) for the G-orbits, and (w, x) for the orthogonal
surfaces:
1
B(w, x)
ds = 2
T (w, x)
2
dw2
dx2
τ1 (x)2
τ2 (w)2
+
± U (w)
+
V
(x)
U (w) V (x)
B(x, y)2
B(x, y)2
,
where B(w, x) (respectively T (w, x)) is a polynomial that is quadratic (respectively
linear) in both w and x, U (w) and V (w) are both quartic polynomials, while τ1 (x)
and τ2 (w) are both quadratic polynomials with values in the cotangent bundle to the
orbits of G satisfying τ1 (x) ∧ τ2 (w) = B(w, x)du ∧ dv.
This similarity in classification results motivates us to look for connections between the two categories where the classifications take place. We find that the Kerr
family of metrics acts as a bridge between these categories. The Kerr metrics have
Lorentzian signature and are used to model isolated rotating black holes. It is well
known in the physics community that if the time and angular momentum variables
for the Kerr metric are chosen to take imaginary rather than real values, then the
metric tensor remains real valued, but has Riemannian signature (see for example
[18]). Such a coordinate transformation is called a Wick rotation. We show that
the Riemannian Kerr metrics which are the result of this Wick rotation admit an
ambitoric structure.
This thesis studies two generalizations of the ambitoric structure of Riemannian
Kerr. The first generalization is the category of regular ambitoric 4-orbifolds. The
main result is a classification of these objects under certain completeness assumptions.
The second generalization comes from the Tomimatsu-Sato family of metrics.
These are a family of Lorentzian stationary and axis-symmetric Ricci-flat metrics
2
which are indexed by a natural number δ. The simplest case, δ = 1, gives the
Kerr metrics. We show that these Tomimatsu-Sato metrics can be Wick-rotated into
Riemannian signature using a straight-forward generalization of the Kerr case. We
then equip the resulting Riemannian metrics with a structure which we call almost
ambitoric, which differs from an ambitoric structure only in the sense that the related
almost complex structures are not required to be integrable. We demonstrate some
properties of the moment maps, with explicit details in the δ = 2.
The introduction continues as follows. We start with a short introduction to
toric geometry. Then there will be a section giving geometric motivation for the
structures that are studied, followed by a summary of our classification results.
1.1
Introduction to toric geometry
We treat toric geometry from a symplectic viewpoint. Let (M, ω) be a symplectic
manifold, and G be a Lie group which acts on M by symplectomorphisms. The action
of G on M is called Hamiltonian if for every X ∈ g with induced vector field XM
on M , XM ω is exact. In other words, there exists some function fX ∈ C ∞ (M )
with dfX = −XM ω. Since the map X → fX is linear, it induces a smooth map
μ : M → g∗ defined by < μ, X >= fX . μ is called the moment map of the action of
G on M .
Definition 1.1.1. A (symplectic) toric manifold (M, ω, T) is a symplectic 2mmanifold equipped with an effective Hamiltonian action of an m-torus T.
A foundational result in studying toric manifolds is the following:
Theorem 1.1.1 ([6]). If (M, ω, T) is a connected compact toric manifold, then μ(M )
is a convex polytope in t∗ .
3
Note that a polytope is simply a generalization of polygons and polyhedra to
arbitrary dimension. An m-dimensional polytope is called simple if m edges meet at
every vertex. A codimension 1 face of a polytope is called a facet. It is clear that an
m-dimensional polytope is simple if and only if each vertex lies at the intersection of
m facets.
Note that the kernel of the exponential map from t to T forms a lattice Λ in t,
called the lattice of circle subgroups. A polytope in t is called rational if every facet
admits a normal which lies in Λ. Such a normal u ∈ Λ to a facet F is called primitive
if for every normal v ∈ Λ to F , there exists some n ∈ Z such that v = nu. It is clear
that each facet F of a rational polytope admits a unique inward-pointing primitive
normal ûF . A simple rational polytope is called Delzant if for each vertex v, written
m
as the intersection of its adjacent facets v = ∩m
i=1 Fi , {ûFi }i=1 forms a basis for Λ.
This property is named after Thomas Delzant, who proves a converse to the
previous theorem:
Theorem 1.1.2 ([12]). If (M, ω, T) is a connected compact toric manifold, then
μ(M ) is a Delzant polytope in t∗ . Conversely, for every Delzant polytope Δ in t∗ ,
there exists a connected compact toric manifold with momentum image Δ which is
unique up to isomorphism.
This theorem has been extended to the orbifold case in [24]. The classification
is in terms of labelled polytopes, which are simple rational polytopes equipped with
a labelling of each facet F by an inward pointing normal uF in Λ. The orbifold
structure group for the moment-map preimage of the interior of a face F is given be
the quotient
Λ∩SpanR {uFi }
,
SpanZ {uFi }
where {Fi } is the set of facets containing F . In particular,
4
the orbifold structure group for the preimage of the interior of a facet F is given by
Z/nF Z, where nF satisfies uF = nF ûF .
Theorem 1.1.3 ([24]). If (M, ω, T) is a connected compact toric orbifold, then μ(M )
is a simple rational polytope in t∗ which admits a labelling determined by the orbifold
structure group of the preimage of the interior of each facet. Conversely, every
labelled polytope arises from some connected compact toric orbifold.
1.2
The Riemannian Goldberg-Sachs theorem and related geometric
structures
Let (M, g) be an oriented Riemannian 4-manifold. We have the usual decom-
position
Λ 2 M = Λ+ M ⊕ Λ − M
of 2-forms into self- and anti-self dual forms. It is well known that the set almost complex structures compatible with g are naturally identified with sections of Λ+ M with
norm equal to 2, via the map J → J . Similarly, one can see that Λ− M corresponds
to compatible almost complex structures which induce the opposite orientation.
Let W be the Weyl curvature tensor for g. Then W is a section of Sym2 (Λ2 M ).
Moreover, there is a decomposition W = W+ + Wi so that W± is a section of
Sym2 (Λ± M ). Using , we can treat W± as a trace-free endomorphism of Λ± M
which is symmetric, and hence diagonalizable. In a Riemannian analogue of the
Petrov classification in Lorentzian signature, there are three algebraic types for W ± ,
corresponding to the number of distinct eigenvalues being 1, 2, or 3:
• W± = 0.
• Λ± M has a unique 1-dimensional eigenspace of W± .
5
• Λ± M splits into 1-dimensional eigenspaces of W± .
In the case that g is Einstein, the Riemannian analogue of the Goldberg-Sachs
theorem allows us to understand which of these almost complex structures is integrable:
Theorem 1.2.1 ([25]). If an orientable Einstein 4-manifold (M, g) admits a compatible complex structure, then Λ2 M does not split into 1-dimensional eigenspaces.
If W+ = 0 = W− , then there are at most two (up to sign) compatible complex structures. If these exist, they correspond to 2-forms lying in the unique 1-dimensional
eigenspaces of W+ and W− .
This motivates the definition of an ambihermitian structure (g, J+ , J− ) to be a
Riemannian metric g with two compatible complex structures J± inducting opposite
orientations. The prefix ambi- is chosen to emphasize the condition of inducing
opposite orientations. Note that by the Riemannian Goldberg-Sachs theorem, if g is
Einstein with W+ = 0 = W− , then the complex structure J± are unique up to sign.
The parallel story in Lorentzian signature was done earlier. In [15], Flaherty
shows that no almost structure can be compatible with a Lorentzian metric. However,
he defines modified almost complex structures to be complex-valued endomorphisms
which square to − Id. These modified almost complex structures can be compatible
with a Lorentzian metric. Moreover, Flaherty shows that if such a metric is Einstein
and has Petrov type D, then locally there exist two unique (up to sign) compatible
modified (integrable) complex structures.
The Kerr metrics are Einstein and type D. Moreover they admit a Wick rotation
to Riemannian signature. The resulting Riemannian Kerr metrics then admit an
6
ambihermitian structure; the modified complex structures guaranteed by Flaherty’s
work become real-valued after Wick rotation. The Kerr family of metrics thus forms
a bridge between the categories of Einstein ambihermitian metrics and Einstein type
D metrics.
Definition 1.2.1. An ambihermitian structure (g, J+ , J− ) is ambikähler if each of
(g, J± ) is conformally Kähler, meaning that there exists some g± ∈ [g] such that
(g± , J± ) is Kähler.
Since ambihermitian structures with Einstein metrics are ambikähler [3], the
Riemannian Kerr metrics admit an ambikähler structure. Moreover, they admit the
action of a 2-torus which is isometrically holomorphic with respect to both Kähler
structures. This torus corresponds to the symmetries of the Lorentzian Kerr metric
being axis-symmetric and stationary, after identifying imaginary time as periodic.
Since all of the above structure depends only on the conformal class of [g], this
motivates the following definition.
Definition 1.2.2. An ambitoric manifold (M, [g], J+ , J− , T) is a 4-manifold M
equipped with
• a conformal class of Riemannian metrics [g],
• two complex structures J± inducing opposite orientations such that there exist
metrics g± ∈ [g] which are Kähler with respect to J± ,
• and the action of a 2-torus T which acts isometrically with respect to both g±
and is Hamiltonian with respect to both Kähler forms.
7
We find that the Riemannian Kerr metrics admit an ambitoric structure. Note
also that we can extend the definition of ambitoric structure to orbifolds without
modification.
1.3
Summary of results
We show that the Riemannian Kerr metrics admit an ambitoric structure. More
generally, we show that the Riemannian Tomimatsu-Sato metrics admit an analogous
structure which we call almost ambitoric. We study the moment maps of these
examples and show that the images are not convex, even locally, near the regions
corresponding to the ring singularities in Lorentzian signature where the curvature
blows up. To understand these moment map images, we use the language of folded
symplectic structures studied in [9]. Folded symplectic structures are 2-forms which
are closed and non-degenerate away from certain hypersurfaces which are called
folding hypersurfaces. We interpret the Kähler forms of the Kerr ambitoric structure
as folded symplectic structures on manifolds which have folding hypersurfaces where
the local convexity of the moment map fails. This result can be understood in the
context of Example 3.11 in [9], which shows that there is no reason for the moment
map to be locally convex near a folding hypersurface.
Next, we discuss our classification results, which will be presented in an upcoming paper [13]. We find it convenient to work with slight generalizations of regular
ambitoric orbifolds which we call regular ambitoric orbifold completions, for which the
full ambitoric structure may only be defined on a dense open subset. More precisely,
we call a Riemannian manifold completable if its Cauchy completion is an orbifold.
A regular ambitoric orbifold completion is defined to be the Cauchy completion of
8
a completable regular ambitoric manifold equipped with a free torus action. Thus
every complete regular ambitoric orbifold is a regular ambitoric orbifold completion,
since the set of its free torus orbits is an open dense submanifold.
Regular ambitoric 4-orbifolds have been locally classified on the set of orbits
where the torus action is free [3]. We use this local classification to generate a set of
examples of regular ambitoric 4-manifolds with free torus action, which we call ambitoric ansatz spaces. These examples include the free orbits of the Riemannian Kerr
examples. These ambitoric ansatz spaces depend on a symmetric quadratic polynomial q(x, y) and two functions of one variable A(x) and B(y), with the manifold
given by
A(q, A, B, T) :=
2
x, y, t ∈ R × T : A(x) > 0, B(y) > 0, q(x, y)(x − y) = 0 ,
equipped with the ambitoric structure given in (1.1).
Note that an ambitoric structure gives a conformal class of metrics. We will
want to study metric properties, so we need to make a choice of metric in this
conformal class. We pick out four distinguished metrics to study, which are unique
up to homothety: the Kähler metrics g± , the barycentric metric g0 , which is an
average of g+ and g− , and a metric which we call gk , which is a generalization of
the Riemannian Kerr metrics. We study gk , since it can be applied to the study of
Killing 2-forms in section 4. Throughout the paper, we often write g1 to be one of
the metrics {g± , g0 , gk }.
9
The first step in our classification is to classify ambitoric ansatz spaces which
are completable with respect to our chosen metric g1 . The strategy taken is to first
compare the g1 -Cauchy completion to the Cauchy completion with respect to a more
convenient metric where the Cauchy completion is easy to compute. This comparison
is done using the tools of Busemann completions, the details of which are provided in
an appendix. We then decompose the convenient boundary into components which
we describe as folds/corners/edges according to their behaviour under the moment
map. We further define a component of the convenient boundary is infinitely distant
if it does not lie in the g1 -Cauchy completion, and a fold is proper if it is not an
edge or a corner. We distinguish between folds being positive/negative in such a way
that proper positive/negative folds are folding hypersurfaces for suitable extensions
of the positive/negative Kähler forms. By studying the g1 -Cauchy completions, we
get the following classification result:
Theorem 1.3.1. Let A be an ambitoric ansatz space and g1 ∈ {g± , g0 , gk }. Then A
is g1 -completable if and only if the following conditions hold:
• A has no proper folds.
• Every edge is either infinitely distant or has a compatible normal (see definition
3.2.4).
• If an edge is a fold but not infinitely distant, then g ∈ {g− , gk }.
• Every corner is infinitely distant unless it is at the intersection of two edges
which are not infinitely distant. If such a corner is a positive (respectively
negative) fold, then g ∈ {g+ , gk } (respectively g ∈ {g− , gk }).
10
Moreover, if A is g1 -completable, then its ambitoric structure extends to AgC1 if and
only if every fold is infinitely distant.
To classify regular ambitoric orbifold completions, we first extend the local classification on the set of free torus orbits to a local classification on the Cauchy completions. This allows us to describe the boundary of the set of free orbits in terms of
folds/edges/corners as was done in the case of ambitoric ansatz spaces. We find that
in the case where there are no proper folds, we can apply a slight modification of the
Lokal-global-Prinzip for convexity theorems [19], which is a tool that can be used to
prove that the moment map of a convex toric orbifold is a convex polytope. This
gives us the following partial classification of regular ambitoric orbifold completions:
Theorem 1.3.2. Let (M, M̊ , g1 , J+ , J− , T) be a connected regular ambitoric orbifold
completion without proper folds, where g1 is one of the special metrics {g0 , g± , gk }.
Then there exists an ambitoric embedding of (M̊ , [g1 ], J+ , J− , T) into some ambitoric
ansatz space without proper folds which is completable with respect to the metric
induced by g1 .
We combine this classification with our classification of completable ambitoric
ansatz spaces to obtain an explicit classification of g0 -complete regular ambitoric
4-orbifolds:
Corollary 1.3.1. A regular ambitoric 4-orbifold which is complete with respect to
the barycentric metric g0 is given (uniquely upto gauge transformation) by the data
of a symmetric quadratic polynomial q(x, y), a pair of intervals
(x− , x+ ), (y− , y+ ) ⊂ R such that (x − y)q(x, y) does not vanish on
(x− , x+ ) × (y− , y+ ), a lattice Λ ⊂ R2 and two smooth positive functions
11
A : (x− , x+ ) → [0, ∞) and B : (y− , y+ ) → [0, ∞) satisfying for all > 0 small
enough:
• If
• If
x± ∓
x±
y± ∓
y±
(x±)
√dx
A(x)
converges, then −2 Ap (x± ) ∈ Λ,
√dy
B(y)
converges, then −2 Bp (y± ) ∈ Λ,
(y±)
• For each α, β ∈ {±}, if the integrals from the previous conditions corresponding
to xα and yβ are both convergent, then (xα − yβ )q(xα , yβ ) = 0,
where p(γ) ∈ t is a convenient choice of normal to the momentum map image of
the level sets {x = γ} and {y = γ}. See definition 3.2.4. Note that similar results
can be obtained for g± , but the third condition which tests the corners would be
more complicated.
As an application for our classification results, we consider compact Riemannian 4-manifolds which admit ∗-Killing 2-forms. These are Riemannian signature
analogues of the Killing-Yano tensor in Lorentzian signature, which describes the
so-called hidden symmetries of the Kerr metric. We build on the work of [17], who
divide Riemannian 4-manifolds admitting non-parallel ∗-Killing 2-forms into 3 types.
They show that one of these types is consists of regular ambitoric orbifold completions, identifying the metric with what we’ve been calling gk . We classify compact
4-manifolds of this type:
Theorem 1.3.3. Let (M, g) be a compact connected oriented regular ambitoric orbifold completion which is a manifold admitting a non-parallel ∗-Killing 2-form. Then
M is diffeomorphic to either S4 , CP2 , or a Hirzebruch surface. Conversely, each of
these manifolds admit a metric with a non-parallel ∗-Killing 2-form.
12
Our techniques can also be applied to obtain a classification of complete Einstein
and constant scalar curvature metrics on regular ambitoric 4-orbifolds, extending the
work on the compact case in [4]. This will be pursued in a follow-up paper.
1.4
Ambitoric structures and ambitoric ansatz spaces
In this section, we introduce some results about ambitoric manifolds from [3].
For an ambitoric manifold, (M, [g], J+ , J− , T), we denote by ω± := g± (J± ·, ·) the
Kähler forms of (M, g± , J± ). Since g± are in the same conformal class, there exists
a positive function f such that g− = f 2 g+ .
Definition 1.4.1. The barycentric metric on an ambiKähler manifold is given by
g0 := f g+ = f −1 g− .
Note that the metrics g± are uniquely chosen within their conformal class up
to homothety, so that f is well-defined up to a multiplicative constant, and g0 is
well-defined up to homothety.
Since each K ∈ t is Hamiltonian with respect to ω± , there exist functions fK± ∈
C ∞ (M ) such that Kω± = −dfK± . The map K → fK± is linear, so it gives an element
of t∗ ⊗ C ∞ (M ). This gives a smooth map μ± : M → t∗ , which is the moment map
for the toric manifold (M, ω± , T).
Ambitoric 4-manifolds come in three families [3], which we will describe in the
rest of the section.
Example 1.4.1. Let (Σ1 , g1 , J1 ) and (Σ2 , g2 , J2 ) be two (Kähler) Riemann surfaces
with non-vanishing Hamiltonian Killing vector fields K1 and K2 respectively. Then
13
Σ1 × Σ2 can be given ambitoric structure
g := g1 ⊕ g2 ,
J± := J1 ⊕ ±J2 ,
t := Span{K1 , K2 }.
Example 1.4.2. Let (M, g, J) be a Kähler surface. For any non-vanishing hamiltonian Killing vector field K, we can define an almost complex structure J− by
⎧
⎪
⎨ J
on Span{K, JK}
J− :=
⎪
⎩ −J on Span{K, JK}⊥
J− has opposite orientation to J. If (M, g, J) is conformally Kähler, then (M, [g], J+ :=
J, J− ) is ambikähler. Such an ambikähler manifold is said to be of Calabi type. In
[2], it is shown that the Kähler quotient of (M, g, J) using the momentum map z of
K is a Riemann surface (Σ, (az − b)gΣ , JΣ ), where a, b are constants, gΣ and JΣ are
a metric and complex structure respectively on Σ. An ambikähler surface of Calabi
type (M, [g], J, J− ) is ambitoric if and only if (Σ, (az −b)gΣ , JΣ ) admits a Hamiltonian
Killing vector field.
For an ambitoric 4-manifold (M, [g], J+ , J− , T), we define tM to be the subset
T M spanned by the Killing vector fields t. On an open dense subset M̊ ⊆ M ,
tM̊ := tM |M̊ is a two dimensional distribution in T M̊ . Since the Killing vector fields
.
t are ω± -Hamiltonian, tM̊ is ω± -Lagrangian, so that J+ tM̊ = J− tM̊ = t⊥
M̊
Since J+ and J− have opposite orientations, they commute. This implies that
the endomorphism −J+ J− of T M is an involution. Thus T M decomposes into ±1
14
eigenbundles of −J+ J− . Let ξM̊ and ηM̊ be the intersections of tM̊ with the +1 and
−1 (respectively) eigenbundles of −J+ J− . Since the eigenbundles are J± invariant,
while tM̊ is not, ξM̊ and ηM̊ must be line bundles.
Let K : t → Γ(tM ) be the function which maps a vector X ∈ t to the associated
Killing vector field on M . Let θ ∈ Ω1 (M, t) be the t-valued one-form defined by the
relations θ ◦ K = Idt and θ|t⊥ = 0. Let
M̊
ξ : M̊ → Pt : p → θ((ξM̊ )p ),
η : M̊ → Pt : p → θ((ηM̊ )p ).
If ξ or η is constant, then its image is the span of some Killing vector field K
which realizes the ambitoric structure as being Calabi type. If neither is constant,
then dξ ∧ dη is non-vanishing [3].
Definition 1.4.2. An ambitoric 4-manifold (M, [g], J+ , J− , T) is regular if dξ ∧ dη is
non-vanishing on an open dense set.
It is clear that every ambitoric structure is either regular or of Calabi type.
We will use extensively this local classification of regular ambitoric surfaces:
Theorem 1.4.1 (Theorem 3 from [3]). Let (M, [g0 ], J+ , J− , T) be a regular ambitoric
4-manifold with barycentric metric g0 and Kähler metrics (g+ , ω+ ) and (g− , ω− ).
Then, about any point in an open dense subset of M , there are t-invariant functions
x, y, a quadratic polynomial q(z) = q0 z 2 + 2q1 z + q2 , smooth sections dτ0 , dτ1 , dτ2 of
15
t∗ , and functions A(z) and B(z) of one variable with respect to which:
dy 2
dx2
+
(1.1)
A(x) B(y)
2
2
2
2
y dτ0 + 2y dτ1 + dτ2
x dτ0 + 2x dτ1 + dτ2
+ A(x)
+ B(y)
,
(x − y)q(x, y)
(x − y)q(x, y)
dx ∧ (y 2 dτ0 + 2y dτ1 + dτ2 ) + dy ∧ (x2 dτ0 + 2x dτ1 + dτ2 )
,
ω+ =
q(x, y)2
dx ∧ (y 2 dτ0 + 2y dτ1 + dτ2 ) − dy ∧ (x2 dτ0 + 2x dτ1 + dτ2 )
ω− =
,
(x − y)2
g0 =
where
2q1 dτ1 = q0 dτ2 + q2 dτ0 ,
q(x, y) = q0 xy + q1 (x + y) + q2 .
Conversely, for any data as above, the above metric and Kähler forms do define an ambitoric Kähler structure on any simply connected open set where ω± are
nondegenerate and g0 is positive definite.
The open dense subset of M where this theorem applies is the maximal open
set M̊ where the Killing vector fields t have maximal rank.
The above theorem motivates us to define a family of examples which we will
call ambitoric ansatz spaces:
A(q, A, B, T) :=
2
x, y, t ∈ R × T : A(x) > 0, B(y) > 0, q(x, y)(x − y) = 0 ,
which we equip with the ambitoric structure given by (1.1) while identifying t with
the infinitesimal vector fields of the action of T.
16
Definition 1.4.3. A map between ambitoric oribifolds is ambitoric if it preserves
all of the ambitoric structure. In particular, it is equivariant under the torus action,
holomorphic with respect to both complex structures, and preserves the conformal
structure.
We will use these to rephrase the above theorem in the case that the toric
structure comes from a Lie group.
Corollary 1.4.1. Let (M, [g0 ], J+ , J− , T) be a regular ambitoric 4-manifold freely
acted on by a 2-torus T. Then for any point p ∈ M , there exists a quadratic polynomial q(z), functions A(z) and B(z), a T-invariant neighbourhood U of p, and an
ambitoric embedding φ : U → A(q, A, B, T).
Proof. Since the action of T is free, t has maximal rank at each point in M . Thus we
can apply the previous theorem to find local coordinates on a neighbourhood U of p
which is naturally identified as a subset of some A(q, A, B, T). Since the ambikähler
structure of M is T-equivariant, these coordinates can naturally be extended to the
orbit U := T · U .
17
CHAPTER 2
Moment maps and folded symplectic structures
2.1
Folded symplectic structures
In this section, we will extend the symplectic structures on ambtoric ansatz
spaces to folded symplectic manifolds, and describe their moment map images.
Definition 2.1.1. [9] A closed 2-form ω on a 2n-dimensional manifold M is a folded
symplectic structure if there exists an embedded hypersurface Z of M such that
ω is non-degenerate on M \Z and ω n = 0 = ω n−1 on Z. Z is called the folding
hypersurface or fold.
Let
F(q, A, B, T) := {(x, y, t1 , t2 ) ∈ R2 × T2 : A(x) > 0, B(y) > 0} ⊇ A(q, A, B, T).
To simplify notation the arguments for F will be tacit. Let Z± := {f ∓1 = 0} ⊆ F
and F± = F\Z∓ .
Proposition 2.1.1. (F± , ω± ) is a folded symplectic manifold with fold Z± |F± , where
ω± is the 2-form defined in (1.1).
Proof. Recall that f =
q(x,y)
.
x−y
It is a routine computation to show that ω± is a closed
non-vanishing 2-form on F± . Also one can compute
2
=
ω±
f ∓2
dx ∧ dc± x ∧ dy ∧ dc± y,
A(x)B(y)
18
2
so that the vanishing locus of ω±
is Z± . Noting that ω± does not vanish along Z±
then tells us that Z± is a folding hypersurface for ω± .
Proposition 2.1.2. μ± (Z± ) is a (possibly degenerate) conic in t∗ , which we will
denote by C± .
Proof. This can be done by an easy direct computation using the normal forms given
in section 2.2. In the hyperbolic case, one finds that
±
1
μ± (Z± ) = {μ±
1 μ2 = − 4 }.
In the elliptic case, one finds that
2
± 2
μ± (Z± ) = {(μ±
1 ) + (μ2 ) = 1}.
In the parabolic case, one finds that
2
+
μ+ (Z+ ) ={(μ+
1 ) = 4μ2 },
μ− (Z− ) ={(0, 12 ), (0, − 12 )}.
Note that the type of the conic matches the type of the ambitoric structure
(aside from the degenerate conic μ− (Z− ) in the parabolic case), so that for example
a hyperbolic ambitoric structure gives a hyperbola as the image of the fold.
The conic (or more precisely, its dual) is used in [4] to study ambitoric compactifications. The key use made in [4] of the conic is the fact that the moment map
sends level sets of x or y to (subsets of) lines tangent to the conic. If the conic is
19
t∗
M/T
Z
C
μ
y
x
Figure 2–1: The moment map near a fold Z which intersects level sets of x and y
where A(x) and B(y) (shown in blue and red respectively).
non-degenerate, this leads to moment map images near the fold that look like the
example shown in figure 2–1. The moment map is a 2 − 1 cover near the conic, with
the image folding along the conic and remaining in the exterior of the conic. This
behaviour was noted at the end of appendix A in [4], so the previous proposition is
not essentially new.
2.2
Normal forms
The results in this section are taken directly from [3]. The classification from
theorem 1.4.1 can be further refined into 3 cases, called parabolic, hyperbolic and
elliptic respectively if the discriminant of q(z) is zero, negative, or positive respectively. Up to homothety, we can assume that the discriminant is 0 or ±1, which is
done for the normal forms described below. We also provide diagrams indicating the
behaviour of the moment maps in the case where A(x) and B(y) are strictly positive
functions.
20
2.2.1
Parabolic type
The parabolic type is characterized by q(z) = 1 and dτ0 = 0. This allows us to
write the ambitoric structure as
g0 =
dy 2
A(x)(dt1 + y dt2 )2 B(y)(dt1 + x dt2 )2
dx2
+
+
+
,
A(x) B(y)
(x − y)2
(x − y)2
ω+ =dx ∧ (dt1 + y dt2 ) + dy ∧ (dt1 + x dt2 ),
ω− =
dx ∧ (dt1 + y dt2 ) dy ∧ (dt1 + x dt2 )
−
.
(x − y)2
(x − y)2
±
∗
the momentum maps μ± = (μ±
1 , μ2 ) : A → t are given by
μ+
1 =x + y,
μ+
2 =xy,
1
,
x−y
x+y
.
μ−
2 =−
2(x − y)
μ−
1 =−
2.2.2
Hyperbolic type
The hyperbolic type is characterized by q(z) = 2z and dτ1 = 0. This allows us
to write the ambitoric structure as
dy 2
A(x)(dt1 + y 2 dt2 )2 B(y)(dt1 + x2 dt2 )2
dx2
+
+
+
,
A(x) B(y)
(x2 − y 2 )2
(x2 − y 2 )2
dx ∧ (dt1 + y 2 dt2 ) dy ∧ (dt1 + x2 dt2 )
ω± =
±
.
(x ± y)2
(x ± y)2
g0 =
21
A/T
Z−
y
x
μ−
μ+
t∗
t∗
y=∞
x=∞
x=y=∞
Figure 2–2: The moment map on a parabolic ambitoric ansatz space.
±
∗
The momentum map μ± = (μ±
1 , μ2 ) : A → t is given by
1
,
x±y
xy
.
μ±
2 =±
x±y
μ±
1 =−
22
A/T
t∗
Z−
μ±
y
Z+
x
{x = ∞} = {y = ∞}
Figure 2–3: The moment map on a hyperbolic ambitoric ansatz space.
2.2.3
Elliptic type
The elliptic type is characterized by q(z) = 1 + z 2 and dτ0 + dτ2 = 0. This allows
us to write the ambitoric structure as
dy 2
A(x)(2y dt1 + (y 2 − 1) dt2 )2 B(y)(2x dt1 + (x2 − 1) dt2 )2
dx2
+
+
+
,
A(x) B(y)
(x − y)2 (1 + xy)2
(x − y)2 (1 + xy)2
dx ∧ (2y dt1 + (y 2 − 1) dt2 ) dy ∧ (2x dt1 + (x2 − 1) dt2 )
ω+ =
+
,
(1 + xy)2
(1 + xy)2
dx ∧ (2y dt1 + (y 2 − 1) dt2 ) dy ∧ (2x dt1 + (x2 − 1) dt2 )
ω− =
−
.
(x − y)2
(x − y)2
g0 =
23
±
∗
The momentum maps μ± = (μ±
1 , μ2 ) : A → t are given by
1 − xy
,
1 + xy
x+y
,
μ+
2 =−
1 + xy
x+y
μ−
,
1 =−
x−y
1 − xy
.
μ−
2 =
x−y
μ+
1 =−
A/T
Z−
Z+
y
x
μ−
μ+
t∗
x=∞
t∗
y=∞
{x = ∞} = {y = ∞}
Figure 2–4: The moment map on an elliptic ambitoric ansatz space.
24
CHAPTER 3
Classification of complete regular ambitoric 4-orbifolds
In this section, we will classify regular ambitoric 4 orbifolds with some completeness assumptions. The idea is to focus on the open dense set where the local
classification, theorem 1.4.1, holds, with the intention of recovering the global structure by studying the asymptotic behaviour. In order to get global results from the
local classification, we will first study how to glue together the local pieces. This is
done by studying the gauge group of regular ambitoric structures.
3.1
Gauge group for ambitoric ansatz spaces
The local form of a regular ambitoric 4-manifold given in Theorem 1.4.1 is not
unique. To study the flexibility in the local form, let U be a neighbourhood of a
point in a regular ambitoric 4-manifold (M, [g0 ], J+ , J− , T) which is small enough so
that the theorem applies to find the data
{x, y, q(z), dτ0 , dτ1 , dτ2 , A(z), B(z)}.
Another application of the theorem could give a different set of data
{x̃, ỹ, q̃(z), dτ̃0 , dτ̃1 , dτ̃2 , Ã(z), B̃(z)}
over U .
Lemma 3.1.1. x̃ is a rational-linear function of x and ỹ is a rational-linear function
of y.
25
Proof. From the way that the coordinates are constructed in theorem 1.4.1, dx and
dx̃ are sections of the line bundle ξ ∗ , so that x̃ is a function of x. Similarly ỹ is a
function of y. From [3], we know that f (x, y) =
q(x,y)
.
x−y
Seeing how f is represented
in these two different coordinates gives
q̃(x̃, ỹ)
q(x, y)
=
.
x−y
x̃ − ỹ
Fix a real number y0 . Then we have that
q̃(x̃, ỹ(y0 ))
q(x, y0 )
.
=
x − y0
x̃ − ỹ(y0 )
The left hand side of this equality is a rational-linear function of x, since q(x, y0 ) is
a linear function of x. Similarly, the right hand side is a rational-linear function of
x̃. Thus we can solve the equation to find that x̃ is a rational-linear function of x.
A similar argument shows that ỹ is rational-linear function of y.
Lemma 3.1.2.
dx̃ dỹ
(x̃ − ỹ)2
,
=
(x − y)2
dx dy
dx̃
q̃(x̃)
=
,
dx
q(x)
dỹ
q̃(ỹ)
=
.
dy
q(y)
Proof. One can compute
1
dx (x − y)(q0 y + q1 ) − q(x, y) + dy (x − y)(q0 x + q1 ) + q(x, y)
df =
(x − y)2
−q̃(ỹ)dx̃ + q̃(x̃)dỹ
−q(y)dx + q(x)dy
=
=
(x − y)2
(x̃ − ỹ)2
Since dx ∧ dx̃ = 0 = dy ∧ dỹ, the above equation can be rearranged to form
dỹ q̃(x̃)
dx̃ q̃(ỹ)
(x̃ − ỹ)2
=
.
=
(x − y)2
dx q(y)
dy q(x)
26
Note that the terms in this equation are expressible as the product of functions of x
and y. Thus
∂2
log
0=
∂x∂y
(x̃ − ỹ)2
(x − y)2
=
dx̃ dỹ
2
2
−
.
2
(x̃ − ỹ) dx dy (x − y)2
The two above equations can be combined to find
dx̃
dx
=
,
q(x)
q̃(x̃)
dy
dỹ
=
.
q(y)
q̃(ỹ)
Lemma 3.1.3. The rational-linear transformations x̃(x) and ỹ(y) are the same, in
the sense that if x̃ =
Proof. Let x̃ =
ax+b
cx+d
ax+b
,
cx+d
then ỹ =
and ỹ =
a y+b
.
c y+d
ay+b
.
cy+d
We compute
a d − b c
dỹ
= .
dy
(c y + d )2
ad − bc
dx̃
=
,
dx
(cx + d)2
From lemma 3.1.2,
(x̃−ỹ)2
(x−y)2
=
dx̃ dỹ
.
dx dy
Plugging in the expressions for x̃(x), ỹ(x) and
their derivatives yields
(ax + b)(c y + d ) − (cx + d)(a y + b )
(x − y)(cx + d)(c y + d )
2
=
(ad − bc)(a d − b c )
.
(cx + d)2 (c y + d )2
Without loss of generality, we may assume that ad − bc and a d − b c are ±1. The
above equation then yields
ad − bc = a d − b c ,
(ax + b)(c y + d ) − (cx + d)(a y + b ) = ±(x − y),
27
which implies
ac = a c,
ad − a d = b c − bc ,
bd = bd .
⎞
⎛
⎛
⎞
b ⎟
⎜a b ⎟
⎠ = ±⎝
⎠, so that ỹ =
d
c d
⎜a
These relations imply that ⎝
c
a y+b
c y+d
=
ay+b
.
cy+d
The above lemmas allow us to identify the gauge group of coordinate transformations which preserve the regular ambitoric
structure
on the set of generic torus
⎡⎛
⎞
⎤
⎢ ⎜a b ⎟⎥
orbits with P SL2 R, where an element ⎣⎝
⎠⎦ ∈ P SL2 R induces the coordinate
c d
transformation x̃ =
ax+b
, ỹ
cx+d
=
ay+b
.
cy+d
Note that the gauge group was already known in
[3] to be P SL2 R by construction, but it is not expressed in local coordinates.
Example 3.1.1. Consider the ambitoric ansatz space
A := A(q, x4 + 1, y 4 + 1, T).
Consider the change of gauge x̃ = − x1 , ỹ = − y1 which transforms A to
Ã := A(q̃, Ã(x̃), B̃(ỹ)).
Since
∂
∂x
and
∂
∂ x̃
span the same line bundle in T A (namely ξ), we can write g0
restricted to this line bundle in both gauges as
g0 |ξ =
dx̃2
dx2
=
.
A(x)
Ã(x̃)
This allows us to compute
Ã(x̃) = A(x)
dx̃
dx
2
4
= (x + 1)
28
−1
x2
2
= x̃4 + 1.
Similarly, one can compute B̃(ỹ) = ỹ 4 + 1. The image of this gauge transformation
is {x̃ỹ = 0} ∩ Ã, but our computations show that Ã(x̃) and B̃(ỹ) extend to smooth
positive functions on {x̃ỹ = 0}. Thus the ambitoric structure extends to the points
in {x̃ỹ = 0} where (x̃ − ỹ)q̃(x̃, ỹ) does not vanish. In terms of the original gauge,
we find that {x = ∞} and {x = −∞} (respectively {y = ∞} and {y = ∞}) glue
together, with the ambitoric structure induced by the inverse gauge transformation
from {x̃ỹ = 0}.
3.2
Asymptotic analysis of ambitoric ansatz spaces
For a regular ambitoric 4-manifold M , there is a dense open subset M̊ where the
torus action is free. Since M̊ is dense, we can study M \M̊ by studying the asymptotics of M̊ with respect to some metric which makes M complete. Theorem 1.4.1
tells us that M̊ locally looks like an ambitoric ansatz space. Since the asymptotic
behaviour is local, understanding the asymptotics of ambitoric ansatz spaces will
allow us to understand the asymptotics of M̊ .
Let U be an open subset of an ambitoric ansatz space A := A(q, A, B, T). We
will study the asymptotics of U by considering its Cauchy completion. However, the
Cauchy completion depends on the choice of metric, and the ambitoric structure gives
a whole conformal class of choices. The metrics that we will study are the barycentric
metric g0 , the Kähler metrics g± = f ∓1 g0 , and the metric gk := (x−y)q(x, y)g0 (which
describes both the Kerr metric and the metric which will show up in section 4). Let
g1 ∈ {g0 , g± , gk }.
29
We will study the Cauchy completion of U with respect to g1 . This will be denoted
by UCg1 , following the notation used in the appendix. We are interested in the case
when U is identified with a subset of a complete orbifold, so the following definition
is natural:
Definition 3.2.1. U is g1 -completable if g1 its g1 -Cauchy completion U is an orbifold
which is Riemannian with respect to a smooth extension of g1 .
We will focus on the case when U = A. We will use the appendix to relate AgC1
to the Cauchy completion with respect to a more convenient metric. In particular,
consider the embedding A → (RP1 )2 × T formed by identifying RP1 as R ∪ {∞} for
each of the x and y coordinate axes in A. Let gf be the flat metric on (RP1 )2 × T.
Since
A=
2
x, y, t ∈ (R) × T : A(x) > 0, B(y) > 0, q(x, y)(x − y) = 0 ,
We find that
g
ACf = {x ∈ RP 1 : A(x) > 0} × {y ∈ RP 1 : B(y) > 0} × T,
where · is the closure in RP1 with the flat topology. We see that ∂ gf A := ACf \A can
g
be decomposed into components of two types: level sets of x or y, or components
of the vanishing locus of (x − y)q(x, y). Since the moment map sends level sets of x
or y to lines, we call these components edges. We call components of the vanishing
locus of (x − y)q(x, y) as folds. We will call a fold negative if f =
it, and positive if
1
f
q(x,y)
x−y
vanishes on
vanishes on it, analogous to the folding hypersurfaces Z± from
section 2.1. A T-orbit where a pair of edges or folds meet is called a corner. A fold
30
which is not an edge or a corner is called a proper fold. Since we are treating corners
separately, when we refer to an edge or a proper fold we will mean its interior and
not include the adjacent corners.
g
Since ACf is compact, by proposition 6.0.2 in the appendix, every point p1 in
g
AgC1 can be represented by a point p2 in ACf , in the sense that there exists a curve
g
in A which converges to p1 in AgC1 and p2 in ACf . Note that this correspondence
g
between p1 ∈ AgC1 and p2 ∈ ACf is not bijective. For example, let’s consider the case
where U = A. If p2 lies on x = ∞, then p2 may be represented by two curves c± (t),
with limt→∞ x ◦ c± (t) = ±∞. Then c± could not both represent the same point in
AgC1 , where x = ∞ and x = −∞ are not identified. As another example, if T does
not act freely at p1 , then there are multiple choices for p2 , since T acts freely on
g
ACf . We will use this correspondence to describe points in UCg1 by the coordinates
g
on ACf , keeping in mind that on UCg1 these may not be coordinate functions and may
be multi-valued.
g
The correspondence between AgC1 and ACf allows us to decompose ∂ g1 A into
components corresponding to the components of ∂ gf A, and we will use the same
language to refer to them: folds, edges, and corners. Note that there may be comg
ponents of ACf which are not represented in AgC1 . We will refer to these components
from the context of g1 as infinitely distant. Formally these lie in the Busemann
completion (see lemma 6.0.6), but we will not dwell on this fact.
We will now see what we can deduce from the assumption that A is g1 completable. We will first find out what makes a component infinitely distant. Then
we will discuss the edges, the folds which aren’t edges, and finally the corners.
31
3.2.1
Folds
We start with a long lemma. This argument is much more involved than the
others in the section.
Lemma 3.2.1. If A is g1 completable, then every proper fold is infinitely distant.
Proof. Let F be a proper fold which is not infinitely distant. The T-orbits in F must
have well-defined g1 -volume. In [4], the g+ volume of the fibre at (x, y) is shown
to be proportional to
A(x)B(y)
.
q(x,y)4
The volumes (upto a constant) of such a fibre with
respect to each of the candidate metrics for g1 are given in the following table:
g1
Volg1 T(x,y)
g+
A(x)B(y)
q(x,y)4
If g1 is g0 , we find that
g0
g−
gk
A(x)B(y)
A(x)B(y)
A(x)B(y)
(x−y)2 q(x,y)2
(x−y)4
A(x)B(y)
must be well-defined on
(x−y)2 q(x,y)2
F . But F is not a
corner or an edge, so A(x) and B(y) are positive functions on F . Thus we must have
that (x − y)2 q(x, y)2 is non-vanishing on F . This contradicts F being a fold, so that
g1 cannot be g0 . Similar arguments show that if F is positive, then g1 = g− , and if
F is negative, then g1 = g+ . It follows that if one writes g1 = φ(x, y)g0 , then φ(x, y)
vanishes on F .
Now we will show that the g1 -distance between T-orbits in F is zero. To see
g
this, let (x1 , y1 , t0 ) and (x2 , y2 , t0 ) be two different points in ACf ∩ F . Consider the
g
curve γ with image in ACf ∩ F ∩ {t = t0 } connecting (x1 , y1 , t0 ) to (x2 , y2 , t0 ). Let Ix
and Iy be the ranges of the functions x and y respectively restricted to the image of
γ. Since F is not a corner or an edge, we can find a δ > 0 small enough such that
A(x) and B(y) are positive functions on the δ-neighbourhoods (with respect to the
metrics induced by gf ) Ixδ and Iyδ of Ix and Iy respectively. We will also choose δ
32
small enough such that neither Ixδ nor Iyδ are dense subsets of RP1 . Consider the set
V := {(x, y, t) ∈ A : x ∈ Ixδ , y ∈ Iyδ , t = t0 }.
Since neither Ixδ nor Iyδ are dense subsets of RP1 , the metric induced by gf on V is
uniformly equivalent to dx2 + dy 2 . The metric induced by g0 on V is
dx2
A(x)
+
dy 2
.
B(y)
Since A(x) and B(y) are both positive on V , g0 is uniformly equivalent to dx2 + dy 2 ,
and hence gf , on V . Thus there exists some C > 0 such that g0 < Cgf on V .
Let > 0. Since φ(x, y) vanishes on F , we can approximate γ with a curve
γ connecting (x1 , y1 , t0 ) to (x2 , y2 , t0 ) with the interior of the image contained in V
satisfying lengthgf (γ) − lengthgf (γ ) < and φ(x, y) ◦ γ < . Since g1 = φ(x, y)g0 ,
we find
dg1 T(x1 ,y1 ) , T(x2 ,y2 ) ≤ lengthg1 (γ ) < lengthg0 (γ )
< C lengthgf (γ ) < C lengthgf (γ) + .
Taking → 0, we find that the g1 -distance between the T-fibres T(x1 ,y1 ) and T(x2 ,y2 )
is zero. This means that F represents only one T-orbit in AgC1 . In order for the
g1 -volume of this orbit to be well-defined, we must have Volg1 (T(x,y) ) constant. In
the case when g1 = g+ , this means that q(x, y)4 ∝ A(x)B(y). But it is only possible
to write q(x, y)4 as a product of a function of x and a function of y in the parabolic
case, where F is an edge. Similarly, we rule out the case where g1 = g− .
If g1 = gk , then A(x) and B(y) are constants, say A and B. We will rule out
this case by a curvature computation. Since (A, g− , ω− ) is Kähler, we have that the
33
anti-self dual part of the Weyl curvature tensor of [g− ] is given by
W− ∝ s− (ω− ⊗ ω− )0− ,
where (·)0 is the projection onto the trace-free part of Sym2 (Λ2 A) and s− is the
scalar curvature of s− . Using gk = (x − y)2 g− , we estimate
W− g−
|s− |
∝
.
2
(x − y)
(x − y)2
Rmgk gk ≥ W− gk =
In [3], s− is computed to be
s− = −
((x − y)2 , A(x))(2) + ((x − y)2 , B(y))(2)
,
(x − y)q(x, y)
where for arbitrary functions f1 (z, w) and f2 (z),
(f1 (z, w), f2 (z))
(2)
=
f12
∂
∂z
∂
f1
∂z
f2
f12
= f1
d2 f2
∂ 2 f1
∂f1 df2
+
6
−
3
f2 .
dz 2
∂z dz
∂z 2
A+B
, so
In particular, since A(x) and B(y) are constants, we find that s− = −2 (x−y)q(x,y)
that Rmgk gk A+B
(x−y)q(x,y)
on A. Since Rmgk gk must be well-defined at the fold
F , where (x − y)q(x, y) vanishes, we must have A + B = 0. This contradicts A(x)
and B(y) being positive functions.
We can use the following lemmas to test the infinitely distant criterion on proper
folds. We first study the barycentric metric g0 :
Lemma 3.2.2. Let g be a metric on a dense set of A. Let F be the vanishing locus
g
of some function φ on ACf . Assume that there exists some r ∈ R such that each
point of F has a neighbourhood where dφg φr . In other words, there exists some
34
C > 0 such that
1 r
φ
C
< dφg < Cφr on the neighbourhood. Then F is infinitely
distant with respect to g if and only if r ≥ 1.
Proof. Let γ : [0, 1) → A be a curve limiting to F . Since limt→1 φ ◦ γ(t) = 0, we can
find some T ∈ [0, 1) such that γ|[T,1) is transverse to the level sets of φ. Thus the
function φ ◦ γ|[T,1) : [T, 1) → R is invertible with image (0, ], where = φ ◦ γ(T ).
Then γ̃ := γ ◦(φ◦γ)−1 : (0, ] → A is a reparametrization of a tail of γ which satisfies
φ ◦ γ̃(τ ) = τ . Taking the derivative of this relation, we find that dφ ◦ γ̃˙ = 1. We can
choose small enough such that the
1 r
φ
C
< dφg < Cφr holds. We compute
∇g φ ˙
˙
γ̃(τ )g dτ ≥
g
, γ̃
lengthg (γ) ≥ lengthg (γ̃) =
∇g φg
0
0
dτ
dτ
dφ ◦ γ̃˙
=
=
dτ .
r
r
dφg γ̃(τ )
0
0 φ ◦ γ̃(τ )
0 τ
dτ
(3.1)
γ̃(τ )
In the case when r ≥ 1, this integral is infinite, so that the g-length of g is infinite.
Since γ is an arbitrary curve limiting to F , we find that F is g-infinitely distant as
claimed.
Conversely, consider the case when r < 1. One can use the g-gradient flow of the
function φ to construct a curve γ limiting to F which satisfies γ̇ = (∇g φ). Clearly γ
admits a parametrization by some parameter τ satisfying φ ◦ γ(τ ) = τ . Let γ̃ be the
restriction of γ to a neighbourhood of limτ →0 γ(τ ) such that dφg φr . Without
loss of generality, we can assume that γ = γ̃. Then (3.1) holds after replacing the
inequalities with equalities. Since r < 1, we find that the g-length of γ is finite, so
that F is not g-infinitely distant.
35
Lemma 3.2.3. If g1 ∈ {g0 , g± , gk }, then all proper folds are not infinitely distant
with respect to g1 .
Proof. First we consider the case of a negative fold F . This is the vanishing locus of
f =
q(x,y)
.
x−y
Up to a change of gauge, we may assume that q(x, y) is not a constant
function. This allows us to identify F as the vanishing locus of q(x, y). It is a simple
computation with the normal forms to verify that dq(x, y) and q(x, y) do not share
any vanishing points. This, combined with the form of g0 from (1.1) allow us to
deduce that dq(x, y)g0 extends to a positive function on F . This allows us to apply
the previous lemma with φ = q(x, y) and r = 0 to deduce that F is not infinitely
1
distant with respect to g0 . Since g± = f ∓1 g0 , dq(x, y)g± = f ± 2 dq(x, y)g0 . Since
(x − y) does not vanish on F , we can then apply the previous lemma with r = ± 12
to deduce that F is not infinitely distant with respect to g± .
Similarly, if 1 = q, then applying the previous lemma with r = 0 gives the
analogous result for gk . If 1 = q, then gk = g+ , so we’ve already seen that F is
not gk -infinitely distant. The case of positive folds follows similarly by switching the
roles of q(x, y) and (x − y).
3.2.2
Edges
We start our investigation of the edges by building criteria to test the infinitely
distance condition. The following lemma lies at the heart of the argument:
Lemma 3.2.4. Let N be a manifold and T > 0. Consider the Riemannian manifold
(N̂ , g) := (0, T ) × N, a(t)2 dt2 + gN (t) ,
36
where a(t) is a smooth positive function on (0, T ) and {gN (t)}t∈(0,T ) is a smooth family
of metrics on N . Then any smooth curve γ : [0, 1) → N̂ satisfying limτ →1 t◦γ(τ ) = 0
has finite g-length if and only if there exist some > 0 such that 0 a(t)dt converges.
Proof. First assume that any curve γ : (0, 1) → M̃ with limτ →0 t ◦ γ(τ ) = 0 has
infinite length. In particular, for any p ∈ N and ∈ (0, T ), the curve γ(τ ) := (τ, p)
has infinite length. But this length is
1
g(γ̇(τ ), γ̇(τ ))dτ =
0
1
a(τ )dτ =
0
a(t)dt,
0
so that this integral cannot converge as claimed.
Conversely, suppose that any curve γ : (0, 1) → M̃ with limτ →0 t ◦ γ(τ ) = 0 has
infinite length. Fix such a gamma. We may reparametrize γ so that for some > 0,
t ◦ γ(τ ) = τ for all τ ∈ (0, ). We then have
g(γ̇(τ ), γ̇(τ ))dτ ≥
g(γ̇(τ ), γ̇(τ ))dτ
∞>
0
0
2
2
≥
a(t) dt (γ̇(τ ), γ̇(τ ))dτ =
a(τ )dτ.
1
0
0
This proves our claim.
We now move on to consider the case g1 = g0 :
Lemma 3.2.5. If an edge E is given by {x = x0 } (respectively {y = y0 }), then E is
infinitely distant with respect to g0 if and only if there does not exist an > 0 such
y +σ dy
x +σ dx
√
√
(respectively y00
) converges. Here σ ∈ {±1} is chosen so
that x00
A(x)
B(y)
that x0 + σ or y0 + σ lies in the range of x or y on A. Also x0 and y0 are assumed
to be finite, which can be arranged after a gauge transformation.
37
Proof. We will prove the case for x0 with σ = 1. Consider a T-orbit O on E. It has
coordinates (x0 , y0 ) for some y0 ∈ R. We can find a neighbourhood Iy of y0 and some
T > 0 such that (x0 , x0 + T ) × Iy × T ⊂ A. Let N = Iy × T. Consider the following
coordinate transformation
φ : M → (0, T ) × N : (x, y, t) → (x − x− , (y, t)).
It is easy to see that φ∗ g0 is of the form a(t)2 dt2 + gN (t), where φ∗ a(t) = √ 1
A(x)
. By
the previous lemma, we find that O is in the g0 -Cauchy completion of M if and only
if there exists some > 0 such that the integral
x− +
a(t)dt =
0
x−
dx
A(x)
converges. Since O was an arbitrary T-orbit on E, this gives the desired result.
To understand how the condition of g1 -completable applies to the edges, we will
borrow from the work of [4]. They define:
Definition 3.2.2. An ambitoric compactification is a compact connected oriented
4-orbifold M with an effective action of 2-torus T such that on the (dense) union
M̊ of the free T-orbits, there is an ambitoci structure (g± , J± , T) for which at least
one of the Kähler metrics extends smoothly to a toric Kähler metric on (M, T). An
ambitoric compactification is regular if the ambitoric structure on M̊ is regular with
(x, y)-coordinates that are globally defined.
We will add some definitions:
38
Definition 3.2.3. A connected component U of an ambitoric ansatz space A is boxtype if all of its folds are also edges or corners. It’s clear that such a U must be of
the form (x− , x+ ) × (y− , y+ ) × T ⊆ A, for some intervals (x− , x+ ), (y− , y+ ) ⊂ R.
Definition 3.2.4. Let E be an edge given by {x = x0 } (respectively {y = y0 }).
We say E has a compatible normal if A(x0 ) = 0 (respectively B(y0 ) = 0) and the
(x0 )
(y0 )
normal vector to μ± (E) given by −2 Ap (x0 ) (respectively 2 Bp (y0 ) ) lies in the lattice
Λ ⊂ t, where for each γ ∈ R, p(γ) ∈ t corresponds to the polynomial pγ (x, y) :=
1
(x−γ)q(y, γ)+ 12 (q(x, γ)(y−γ)
2
under the identification of t with symmetric quadratic
polynomials orthogonal to q(x, y). See [3] for details on this identification.
Note that the above definition is given for the case when x0 or y0 is finite. In the
infinite case, one can reduce to the finite case by performing a gauge transformation
(for example z → − z1 ).
We can then paraphrase the classification of ambitoric compactifications (proposition 3 in [4]):
Proposition 3.2.1. A box-type component of an ambitoric ansatz space is the interior of an ambitoric compactification if and only if each of its edges has a compatible
normal.
This will allow us to prove:
Lemma 3.2.6. If A is g1 -completable, then each edge is either infinitely distant, a
fold, or has compatible normal.
Proof. Let E be an edge which is neither a fold nor infinitely distant. Since A is is
g1 -completable, E lies in AgC1 as a set of points with at worst orbifold singularities,
and g1 extends to E as a smooth metric. Let φ(x, y) be the function such that
39
g+ = φ(x, y)g1 . Since E is not a fold, φ(x, y) is a positive function along E. This
implies that g+ also extends to E as a smooth metric.
We want to apply the previous proposition here. The proof of that proposition
is local in nature, so we can apply it locally. More explicitly, if both g+ and ω+
extend to E, then E looks like a piece of an ambitoric compactification, so we can
deduce that E has a compatible normal. We have shown that g+ extends, but not
yet ω+ . However, the proof of the previous proposition does not actually require that
ω+ extends, although it certainly is necessary by the converse. To see this, let p be a
point on E. Without loss of generality, by switching the roles of x and y if necessary,
E is of the form {x = x0 } and the T-orbit T · p is given by {x = x0 , y = y0 }. Let
Ay=y0 = {(x, y, t) ∈ A : y = y0 }. Then the normal bundle to T · p in Tp Ay=y0 is a
vector space which coincides with the space Vp in the proof of proposition 1 of [5].
The construction of Vp was the only part of that direction of that proof which used
the symplectic structure, so we can use it to deduce boundary conditions which are
shown in [4] to be equivalent to the condition that E has a compatible normal.
Lemma 3.2.7. If A is g1 -completable, then every edge which is also a fold but not
infinitely distant must have a compatible normal and g1 ∈ {g− , gk }.
Proof. Let E be such an edge. An edge can only be a fold in the parabolic case,
and then fold must be negative. We will change the gauge so that q(x, y) = xy, so
that the E is not {x = ∞} or {y = ∞}. We consider the case when E is given by
{x = 0}, while the case {y = 0} can be treated similarly.
To rule out g1 ∈ {g0 , g+ }, we use the fact that the volume of a T-fibre on E
must be well-defined, as in lemma 3.2.1. First consider the case g1 = g0 . We saw in
40
lemma 3.2.1 that the g0 volume of the T-fibre at (x, y) is given by
A(x)B(y)
.
(x−y)2 x2 y 2
For this
to be defined at x = 0, A(x) must vanish to at least second order at 0. By lemma
3.2.5, this implies that E is infinitely distant with respect to g0 , contradicting the
definition of E.
Similarly for g1 = g+ , we find that
A(x)B(y)
x4 y 4
must be well-defined, implying that
A(x) vanishes to order 4 at 0. A similar argument to lemma 3.2.5 shows that E
is infinitely distant with respect to g+ if and only if there does not exist an > 0
such that the integral 0 √xdx converges. This is the case when A(x) vanishes to
A(x)
order 4 at 0, contradicting the definition of E as not infinitely distant with respect
to g1 = g+ .
The fact that the edge must have a compatible normal follows the same argument
as the previous lemma, replacing g+ and ω+ with g− and ω− .
3.2.3
Corners
Lemma 3.2.8. If A is g1 -completable, then every corner where an infinitely distant
edge or fold meets another edge or fold must be infinitely distant.
Proof. Assume that two infinitely distant edges or folds meet at a corner C which is
not infinitely distant. We can find a small neighbourhood of this corner of the form
((C\{0}) × (C\{0})) ∪ C,
where C is glued into (0, 0) as a subgroup of T. This is never an orbifold, contradicting A being g1 -completable.
Similarly, let C be a corner where an infinitely distant edge or fold E∞ meets
an edge or fold Ef which is not infinitely distant. By lemmas 3.2.1, 3.2.6 and 3.2.7,
41
Ef must be an edge with a compatible normal. It follows that points of Ef have
neighbourhoods of the form C/Γ × C, where Γ is the orbifold covering group. It
follows that C has a neighbourhood of the form
((C/Γ) × (C\{0})) ∪ C,
where C is glued into (0, 0) as a subgroup of T. This is never an orbifold, contradicting A being g1 -completable.
Lemma 3.2.9. If A is g1 completable and C is a corner at the intersection of two
edges which are not infinitely distant, then if C is a positive fold, then g1 ∈ {g+ , gk },
while if C is a negative fold, then g1 ∈ {g− , gk }.
Proof. We will prove the case that C is a positive fold. The case when C is a negative
fold is treated similarly. By lemmas 3.2.6 and 3.2.7, the edges adjacent to C must
have compatible normals. This implies that A(x) and B(y) both vanish to order 1
at C. For topological reasons, C cannot be a free T fibre with respect to g1 . In
particular, the volume of the fibre at C must vanish. Again we refer to the table of
fibre volumes in lemma 3.2.1 to deduce that g1 ∈ {g+ , gk }.
Now we can combine all of our asymptotic analysis to prove our classification of
completable ambitoric ansatz spaces:
Proof of theorem 1.3.1. First assume that A is g1 completable. By lemma 3.2.1,
every proper fold must be infinitely distant. But by lemma 3.2.3, no proper fold is
infinitely distant. Thus A can have no proper folds, which is the first statement.
The next two statements are the result of lemmas 3.2.6 and 3.2.7.
42
By lemma 3.2.9, every corner is infinitely distant unless it is at the intersection
of two folds or edges which are not infinitely distant. However, we’ve seen that a fold
or an edge which is not infinitely distant must be an edge with a compatible normal.
This added to lemma 3.2.9 gives the last condition.
Conversely, assume that the stated conditions hold. The first condition combined with lemma 3.2.3 show that each fold is either infinitely distant or an edge.
The second condition implies that each edge which is not infinitely distant has
a compatible normal. Let E be such an edge. Since the proof of proposition 3.2.1 is
g
local in nature, it allows us to deduce that for one of the metrics g± , E lies in AC± as
a set of points with at worst orbifold singularities, and g± extends to E as a smooth
metric. In the case that E is not a fold, then the conformal factor relating g1 to g±
extends to a positive function on E, so that g1 also extends to E as a smooth metric.
If E is a fold, then the choice of g± must have been g− by the fourth condition. The
fourth condition also ensures that g1 = η(x, y)g− , for some η(x, y) ∈ {1, (x − y)2 }.
Since η(x, y) is positive on E, g1 extends to E as a smooth metric.
Finally, let C be a corner which is not infinitely distant. By the last condition,
C must lie at the intersection of two edges which are not infinitely distant, and
thus have compatible normals. As in the previous paragraph, we can locally apply
g
proposition 3.2.1 to deduce that for one of g± , C lies in AC± as a point with at
worst an orbifold singularity. The conditions that we put on g1 if C is a fold ensures
that the conformal factor relating g1 to g± on A extends to C as a positive function,
ensuring that g1 extends smoothly to C.
43
We’ve shown that every point in AgC1 has at worst orbifold singularities, and that
g1 extends to a smooth metric on AgC1 . Thus A is g1 -completable as claimed.
To prove the moreover, first note that if the ambitoric structure extends to AgC1 ,
then the function f is a conformal factor between the metrics g+ and g0 on AgC1 . In
particular, f must be a smooth positive function on AgC1 . It follows that AgC1 has no
folds. In other words, every fold is infinitely distant.
Conversely, if every fold is infinitely distant, then the functions q(x, y) and (x−y)
extend to smooth positive functions on AgC1 . We can use these functions to construct
the conformal factor between g± and g− on AgC1 , so that g± both extend as smooth
metrics to AgC1 . Similarly, by considering the expressions for ω± in (1.1), these also
extend to AgC1 , so that AgC1 is ambitoric.
3.3
Classification of regular ambitoric orbifold completions
In this section, we will use the results of the previous section to study regular
ambitoric orbifold completions (recall that these are defined in the introduction). In
the following section, we will extend the local classification of the set of free orbits of
regular ambitoric orbifold completion (corollary 1.4.1) to a local classification on the
completion. In the following section, we modify the tool of the Lokal-global-Prinzip
for convexity theorems [19]. Finally, we show how this modified Lokal-global-Prinzip
can be applied to get classification results for regular ambitoric orbifold completions.
3.3.1
Local asymptotics
We will first work with local embeddings. Recall corollary 1.4.1:
44
Corollary 3.3.1. Let (M, [g0 ], J+ , J− , T) be a regular ambitoric 4-manifold freely
acted on by a 2-torus T. Then for any point p ∈ M , there exists a quadratic polynomial q(z), functions A(z) and B(z), a T-invariant neighbourhood U of p, and an
ambitoric embedding φ : U → A(q, A, B, T).
We can use the following lemma to glue these local embeddings together:
Lemma 3.3.1. Let (M, [g0 ], J+ , J− , T) be a regular ambitoric 4-manifold with a free
T-action such that M/T is contractible. Then there exists a T-equivariant local embedding φ : M → (RP1 )2 × T such that for any p ∈ M , there exists an ambitoric
embedding φp : Up → Ap of a neighbourhood Up of p into an ambitoric ansatz space
Ap such that the following diagram commutes:
φ
MO
?
Up /
2
(RP 1 )O × T

/
φp
?
Ap
Moreover, if φ is injective and for each z ∈ RP1 , φ(M ) ∩ {z} × RP1 × T and
φ(M ) ∩ RP1 × {z} × T are connected or empty, then φ is an ambitoric embedding
into some ambitoric ansatz space A.
Proof. From corollary 1.4.1, M is covered by T-invariant charts {φα : Uα → Aα }α
from M into ambitoric ansatz spaces {Aα }α . From section 3.1, if Uα ∩ Uβ = ∅, then
φβ ◦ φ−1
α : φα (Uα ∩ Uβ ) → φβ (Uα ∩ Uβ )
is given by some element of P SL2 R. Thus the maps {φα }α induce a C̆ech co-cycle
C in H 1 (M, P SL2 R). Since the φα are T-equivariant, C is T invariant. Thus C is
45
equivalent to a C̆ech co-cycle C̄ in H 1 (M/T, P SL2 R). Since M/T is contractible, C̄,
and hence C, must be trivial. This means that the gauge can be chosen globally, so
that the φα s can be glued to form a map φ : M → (RP1 )2 × T.
Now let p ∈ M . Then p ∈ Uα for some alpha. Transforming the gauge of
Uα → Aα to match the global gauge used to construct φ results in an embedding
Up → Ap which has the desired properties by construction.
To prove the moreover, assume that φ has the required properties. We need
to show that there exist functions A(x), B(y) and q(x, y) on the image of φ which
agree with those given be the local embeddings {φp }p∈M . For each x0 ∈ RP1 , since
φ(M ) ∩ {x0 } × RP1 × T is connected, the {φp }p∈M must agree on the value of
A(x0 ). Since x0 was arbitrary, it follows that A(x) is uniquely determined on φ(M ).
Similarly B(y) is uniquely determined on φ(M ). The fact that q(x, y) is uniquely
determined follows from the fact that it is a quadratic function, thus determined on
a connected set by its value on any open subset.
We will have to work a little harder to get asymptotic information from our
local ambitoric charts.
Let (M, [g0 ], J+ , J− , T) be a regular ambitoric 4-manifold with free T-orbits. Let
{λ1 , λ2 } be a set of generators of Λ.
λ21 + λ22 ∈ Sym2 (t) ∼
= Sym2 (t∗ )∗
is a Riemannian metric on t∗ , identifying t∗ with Euclidean space. The momentum
map μ+ : M → t∗ is T-invariant, so it descends to an orbital momentum map
μ̄+ : M/T → t∗ , which is a local embedding. Thus ḡE := (μ̄+ )∗ (λ21 + λ22 ) is a
46
Riemannian metric on M/T. Since t is the tangent space to T, (λ21 + λ22 )−1 is a
Riemannian metric on T. Thus
gE := (μ+ )∗ (λ21 + λ22 ) + (λ21 + λ22 )−1
is a T-invariant metric on M which descends to the metric ḡE on M/T. gE will be
used in the same way at the flat metric gf was used for ambitoric ansatz spaces.
Lemma 3.3.2. Let (M, M̊ , g1 , J+ , J− , T) be a regular ambitoric orbifold completion.
Then every point p in ∂ gE M̊ has a neighbourhood in M̊CgE which can be identified with
g
an open set in ACf for some ambitoric ansatz space A such that p is identified with
a point in ∂ gf A.
Proof. We first treat the case when T · p admits a neighbourhood U in M̊CgE /T such
that U ∩ (M̊ /T) is contractible. Let Û ⊆ M̊ be the union of the orbits U ∩ (M̊ /T).
By lemma 3.3.1, by shrinking U if necessary, we can find an ambitoric embedding
φ : Û → A of Û into some ambitoric ansatz space A. It is easy to see that φ∗ gf
induces the same topology as gE on ÛCgE . Thus we find that φ naturally extends to
g
an injection φ̄ : ÛCgE → ACf . Since p ∈ ∂ gE Û , we have φ̄(p) ∈ ∂ gf φ(Û ).
If φ̄(p) ∈
/ ∂ gf A, then φ̄(p) ∈ A. Since g1 |M̊ is homothetic to one of the special
metrics {g0 , g± , gk } of the ambitoric structure on M̊ , φ∗ (g1 |M̊ ) must be homothetic
to one of the special metrics of the ambitoric structure on φ(Û ) ⊆ A. Thus φ∗ (g1 |M̊ )
extends to a complete metric on A, implying that
φ̄(p) ∈ ∂ φ∗ (g1 |M̊ ) φ(Û ).
Moreover, since Û is T-invariant, so is φ(Û ), and thus
∂ φ∗ (g1 |M̊ ) φ(Û ). Thus the free orbit T · φ̄(p) lies in ∂ φ∗ (g1 |M̊ ) φ(Û ). Reversing the
correspondence shows that there is a free T-orbit O ⊂ ∂ g1 Û corresponding to the
47
orbit T · p ⊂ ∂ gE Û via lemma 6.0.2. Since O is a free orbit, it must lie in M̊ by
definition. But on M̊ , the topologies induced by gE and g1 are equivalent. Thus
p ∈ M̊ . This contradicts p ∈ ∂ gE M̊ , so that φ̄(p) ∈ ∂ gf A as claimed.
We now treat the case where every neighbourhood U of T · p satisfies the condition that U ∩ (M̊ /T) is not contractible. We will show that this case is not possible,
essentially because for an ambitoric ansatz space A, ∂ gf A does not have isolated
T-orbits. Since MCgE /T is 2-dimensional, we can find a neighbourhood U of T · p such
that U ∩ (M̊ /T) is homeomorphic to a punctured disc. Let V ⊂ U ∩ (M̊ /T) be a
contractible set obtained by cutting a line L between the two boundary components
of the puctured disc U ∩ (M̊ /T). Let V̂ ⊆ M̊ be the union of T-orbits V . We can
then use lemma 3.3.1 to get a local embedding φ : V̂ → (RP1 )2 × T. Arguing as in
the previous case, φ extends to a local embedding φ̄ : V̂CgE → (RP1 )2 × T. Still using
lemma 3.3.1, Since V̂CgE is compact, we can cover V̂ with finitely many {V̂α }α such
that each φ|V̂α is an embedding into some ambitoric ansatz space Aα ⊂ (RP1 )2 × T.
Let S be the set of α such that p ∈ ∂ gE V̂α . We can repeat the argument from the
previous case for each α to deduce that φ̄(p) ∈ ∂ gf Aα for each α ∈ S. But the
{∂ gf Aα }α are (possibly singular) hypersurfaces which must locally agree, so there
must be some neighbourhood W of φ̄(p) and some hypersurface H in W passing
through φ̄(p) which such that ∂ gf Aα ∩ W = H. We treat the case where H is not
singular at φ̄(p) for clarity. By shrinking V small enough so that φ̄(V ) ⊂ W we find
that φ(V̂ ) lies on one side of H in W . It follows that the ray in TT·p (VCḡE ) corresponding to L must cover the line (Tφ̄(p) H)/t ⊂ TT·φ̄(p) (RP1 )2 by the map induced by
φ̄. This is impossible, since a linear map cannot send a ray to a line.
48
We use the previous lemma to decompose ∂ gE M̊ into components corresponding to the decomposition of ∂ gf A into folds, edges, and corners. We will use this
language to discuss the components of ∂ gE M̊ . As in the previous section, we will
use proposition 6.0.2 to decompose ∂ g1 M̊ into components, which we will call folds,
edges, and corners likewise. We call components of ∂ gE M̊ which are not in ∂ g1 M̊
infinitely distant.
The following example is a variation on example 2.8 from [23], and shows that
the orbital momentum map is not always a global embedding:
Example 3.3.1. Consider the elliptic type ambitoric ansatz space
A := A xy + 1, x4 + 1, y 4 + 1, T .
Consider the space Ā obtained by gluing the edges {x = ∞} and {x = −∞} together,
as well as the edges {y = ∞} and {y = −∞} together. As in example 3.1.1, the
ambitoric structure from A extends to Ā. Since the functions A(x) and B(y) are
positive on Ā, Ā has no edges. It follows that Ā has two connected components. Let
Ā0 be one of the connected components of Ā.
From section 2.1, we find that the moment map μ+ : Ā0 → t∗ has an image Δ+
which is the exterior of a conic C+ . Since A is elliptic type, C+ is an ellipse. Moreover,
the μ+ -fibre over any point in Δ+ is a single T-orbit in Ā0 , allowing us to identify
Ā0 /T ∼
= Δ+ . Since Δ+ is not simply connected, its universal cover π : Δ̃+ → Δ+ is
not injective. Since μ+ : Ā0 → Δ+ is a principal T-bundle, we can use π to construct
the pull-back bundle Ã := π ∗ Ā0 over Δ̃+ . Since π is a covering map, it extends to
49
a covering map from Ã to Ā0 , which can be used to induce the ambitoric structure
from Ā0 to Ã.
We find that Ã is an ambitoric manifold with orbit space Ã/T ∼
= Δ̃+ and orbital
moment map π : Δ̃+ → Δ+ ⊂ t∗ which is not injective.
The main feature of the previous example was the existence of a proper fold. This
motivates us to consider only regular ambitoric orbifold completions without proper
folds, since we want to rule out the possibility that the orbital moment map is not
injective. If (M, M̊ , g1 , J+ , J− , T) is a regular ambitoric orbifold completion without
proper folds, then the boundary ∂ gE M̊ consists of only edges and the corners where
ḡE
→ t∗ ,
they meet. The orbital moment map naturally extends to a map (M̊ /T)C
which is locally convex since ∂ gE M̊ consists of only edges and corners, which each
get locally sent to lines and convex cones respectively. As in the case of compact
toric orbifolds, we wish to use this local convexity to prove global convexity, which
will give us control over the topology of M . The tool to do this is the Lokal-globalPrinzip for convexity theorems [19], although it requires that the momentum map is
ḡE
proper. Since (M̊ /T)C
may not be compact a priori, we have to consider the case
that the momentum map is not proper. However, in the next section we will see that
completeness is actually enough for the proof of Lokal-global-Prinzip to hold.
3.3.2
Lokal-global-Prinzip
We begin this section by recalling some definitions from [19], and then stating
their version of the Lokal-global-Prinzip. We will then state and prove the slight
modification of this result which we will need. Note that it is likely that what we
prove here is only a case of the general treatment of the Lokal-global-Prinzip in [28]
50
(where we got the idea to use Hopf-Rinow), but for our purposes we find it convenient
to prove the result.
Definition 3.3.1. A continuous map Ψ : X → V from a connected Hausdorff
topological space X to a finite dimensional vector space V is locally fibre connected if
every point in X admits an arbitrarily small neighbourhood U such that Ψ−1 (Ψ(u))∩
U is connected for each u ∈ U .
Definition 3.3.2. A locally fibre connected map Ψ : X → V has local convexity data
if every x ∈ X admits an arbitrarily small neighbourhood Ux and a closed convex
cone Cx ⊆ V with vertex Ψ(x) satisfying:
• Ψ(Ux ) is a neighbourhood of Ψ(x) in Cx ,
• Ψ|Ux : Ux → Cx is open,
• Ψ−1 (Ψ(u)) ∩ Ux is connected for each u ∈ Ux ,
where the topology on Cx is the subspace topology induced from V .
These definitions allow us to state the theorem:
Theorem 3.3.1 (Lokal-global-Prinzip for convexity theorems [19]). Let Ψ : X → V
be a proper locally fibre connected map with local convexity data. Then Ψ(X) is a
closed locally polyhedral convex subset of V , the fibres Ψ−1 (v) are all connected, and
Ψ : X → Ψ(X) is a open mapping.
Since we are essentially reproducing this theorem, we need to continue defining
the constructions used in the proof.
Consider the equivalence relation on X defined by x ∼ y if and only if x and y
are both contained in the same connected component of a fibre of Ψ. The quotient
51
space X̃ := X/ ∼ is called the Ψ-quotient of X. Let Ψ̃ : X̃ → V be the map induced
by Ψ on X̃.
Let dV be the usual Euclidean distance on V with respect to some fixed basis.
We can use this to define a distance on X̃ as follows:
d : X̃ × X̃ → [0, ∞) : (x̃, ỹ) →
inf
γ∈Γ(x̃,ỹ)
lengthdV (Ψ̃ ◦ γ),
where Γ(x̃, ỹ) is the set of curves γ connecting x̃ to ỹ such that Ψ ◦ γ is piecewise
differentiable. d is called the metric induced on X̃ by Ψ̃.
Now we can prove our version of the Lokal-global Prinzip for convexity theorems.
Note that the proof is the same as the relevant parts of the previous theorem aside
from the replacement of the use of properness with an appeal to the Hopf-Rinow
theorem.
Theorem 3.3.2. Let Ψ : X → V be a locally fibre connected map with local convexity
data. Moreover, assume that X̃ is a complete locally compact length space with respect
to the metric induced by Ψ̃. Then φ(X) is a convex subset of V and the fibres Ψ−1 (v)
are all connected.
Proof. Let x̃0 , x̃1 ∈ X̃ with c := d(x̃0 , x̃1 ). For each n ∈ N, we can find some
(n)
γn ∈ Γ(x̃0 , x̃1 ) such that lengthdV (Ψ̃ ◦ γ) ≤ c + n1 . For each n ∈ N, let x̃ 1 be the
2
∞
(n)
is a sequence in the closed ball Bc+1 (x̃0 ) of radius
midpoint of γn . Then x̃ 1
2
n=1
c + 1 about x̃0 . Since X̃ is a complete locally compact length space, the Hopf-Rinow
∞
(n)
admits a subsequence
theorem [8] tells us that Bc+1 (x̃0 ) is compact. Thus x̃ 1
2
52
n=1
which converges to some x̃ 1 ∈ Bc+1 (x̃0 ), which must satisfy
2
c
d x̃0 , x̃ 1 = d x̃ 1 , x̃1 = .
2
2
2
Repeating the argument for the pairs of points (x̃0 , x̃ 1 ) and (x̃ 1 , x̃1 ), we construct
2
2
points x̃ 1 and x̃ 3 satisfying
4
4
n
p d x̃ 2nm , x̃ 2pq = c m − q 2
2
(3.2)
for all suitable n, m, p, q. Inductively we can repeat the argument to construct points
x̃ 2nm for n, m ∈ N with 0 ≤ n ≤ 2m satisfying (3.2). We can then extend the function
n
→ x̃ 2nm to a continuous function γ : [0, 1] → X̃ satisfying d γ(t), γ(t ) = c|t − t |
2m
for all t, t ∈ [0, 1]. This means that locally
dV Ψ̃ ◦ γ(t), Ψ̃ ◦ γ(t ) = c|t − t |,
which can only happen if Ψ̃ ◦ γ is a straight line segment. This line segment connects
Ψ̃(x̃0 ) and Ψ̃(x̃1 ). Varying x̃0 and x̃1 , we find that Ψ(X) = Ψ̃(X̃) is convex.
To prove that the fibres are all connected, let x̃0 , x̃1 ∈ X̃ such that Ψ̃(x̃0 ) =
Ψ̃(x̃1 ) =: v. Let γ : [0, 1] → X̃ be the curve constructed above. Then Ψ̃ ◦ γ is a
line segment containing v. Assume that Ψ̃ ◦ γ is not the constant function v. Then
Ψ̃ ◦ γ is a loop through v as well as a straight line segment. Thus there exists a
turning point v0 = Ψ̃ ◦ γ(t0 ) such that Ψ̃ ◦ γ [t0 − , t0 ] = Ψ̃ ◦ γ [t0 , t0 + ] for all
> 0 sufficiently small. Thus Ψ̃ is not locally injective near γ(t0 ). This contradicts
Ψ being locally fibre connected. Thus Ψ̃ ◦ γ is the constant function v. This implies
that d(x̃0 , x̃1 ) = 0, so that x̃0 = x̃1 as required.
53
We recall proposition 3.25 from [8]:
Proposition 3.3.1. Let X be a length space and X̃ be a Hausdorff topological space.
Let p : X̃ → X be a continuous local homeomorphism, and d be the metric induced
on X̃ by p.
1. If one endows X̃ with the metric d, then p becomes a local isometry.
2. d is a length metric.
3. d is the unique metric on X̃ that satisfies properties (1) and (2).
Combining this proposition with the previous theorem gives:
Corollary 3.3.2. Let Ψ : X → V be a locally fibre connected map with local convexity
data. Moreover, assume that Ψ̃ is a local homeomorphism, and that X̃ is complete
with respect to the metric d˜ given by the pull-back with respect to Ψ̃ of the Euclidean
metric on V . Then φ(X) is a convex subset of V and the fibres Ψ−1 (v) are all
connected.
Proof. By the previous theorem, it suffices to show that (X̃, d) is a complete locally
compact length space. Since Ψ̃ : X̃ → V is a local homeomorphism and V is locally
compact (being a vector space), X̃ is locally compact. By the previous proposition,
(X̃, d) is a length space. Note that d˜ and d both satisfy properties (1) and (2) in
˜ Since (X̃, d)
˜ is
the previous proposition. Thus by uniqueness, we have that d = d.
complete, (X̃, d) is complete.
3.3.3
Classification results
We can now combine the results of the previous two sections to obtain the main
classification results. It is clear from the definitions that the Cauchy completion of a
completable ambitoric ansatz space is a regular ambitoric orbifold completion. The
54
following theorem gives a converse in the case that there are no proper folds. Note
that this can be combined with our classification of completable ambitoric ansatz
spaces to give a classification of regular ambitoric orbifold completion without folds.
The following lemma will be the key which allows us to use our Lokal-globalPrinzip:
Lemma 3.3.3. Let A0 be a connected component of some ambitoric ansatz space
without proper folds. Then the orbital moment map μ̄+ : A0 /T → t∗ extends to
ḡE
as a local homeomorphism with local convexity data.
(A0 /T)C
Proof. μ̄+ is a rational map from A0 /T ⊂ (RP1 )2 to t∗ . Thus μ̄+ extends naturally
ḡE
⊂ (RP1 )2 , which we will also denote by
to a map on (RP1 )2 , and hence on (A0 /T)C
μ̄+ . Since A0 /T is connected and μ̄+ |A0 /T is a homeomorphism, so is μ̄+ |(A0 /T)ḡE .
C
ḡE
Since there are no proper folds, ∂ (A0 /T) consists of finitely many edges
which μ̄+ sends to lines tangent to the conic C+ . We find that for every corner
c ∈ ∂ ḡE (A0 /T), arbitrarily small neighbourhoods of c get mapped by μ̄+ to neighbourhoods of μ̄+ (c) in the convex cone with vertex μ̄+ (c) formed by the rays passing
through C+ . By similarly describing neighbourhoods of points on edges and in A0 ,
ḡE
→ t∗ has local convexity data.
we find that μ̄+ : (A0 /T)C
We can now prove our classification of regular ambitoric orbifold completions:
ḡE
Proof of theorem 1.3.2. By lemma 3.3.2, (M̊ /T)C
is locally covered by charts with
g
image in (A/T)Cf . Applying the previous lemma to these local charts, we find that
ḡE
μ̄+ naturally extends to a local homeomorphism μ̄+ : (M̊ /T)C
→ t∗ with local
convexity data.
55
Since ḡE is the pull-back of a Euclidean metric on t∗ by μ̄+ , we can apply corollary
ḡE
3.3.2 with X = X̃ = (M̊ /T)C
, V = t∗ and Ψ = Ψ̃ = μ̄+ . This tells us that μ̄+ is
injective with convex image. In particular M̊ /T is contractible. We can then apply
lemma 3.3.1 to construct a T-equivariant local embedding φ : M̊ → (RP1 )2 × T. Let
φ̄ : M̊ /T → (RP1 )2 be the map induced by φ. We have the following commutative
diagram:
RP1
2
O
×T
//
φ
RP1
2
μ̄+
.
μ̄+
O
φ̄
/ t∗
=
/ / M̊ /T
M̊
It follows that φ̄ is injective. Thus φ is injective. It also follows that since
μ+ (M̊ ) is convex, φ(M̊ ) intersects each level set of x or y in at most one connected
component. We are now able to apply the full force of lemma 3.3.1 to deduce that
the image of φ can be given the structure of an ambitoric ansatz space. Since M̊ has
no proper folds, φ(M ) is box-type.
Corollary 3.3.3. Let (M, M̊ , g1 , J+ , J− , T) be a regular ambitoric orbifold completion where g1 is either g0 or gk . Then there exists an ambitoric embedding of
(M̊ , [g1 ], J+ , J− , T) into some box-type ambitoric ansatz space which is completable
with respect to the metric induced by g1 .
Proof. By the previous theorem, it suffices to show that M has no proper folds. This
follows from theorem 1.3.1, since we are not considering the cases when g1 = g± .
We are now in a position to prove our classification of regular ambitoric 4orbfolds:
56
Proof of corollary 1.3.1. A regular ambitoric 4-orbifold M which is complete with
respect to g0 is a regular ambitoric orbifold completion with respect to g0 . We can
then apply theorem 1.3.2 to embed the set of free orbits M̊ of M in the Cauchy
completion (with respect g0 ) of an ambitoric ansatz space A(q, A, B, T) which has
polygonal moment map images. Moreover, since A is completable, it has no proper
folds by theorem 1.3.1. Combining these facts, we find that A is of box-type. In particular there exist x± , y± ∈ R such that M̊ ∼
= (x− , x+ ) × (y− , y+ ) × T ⊂ A(q, A, B, T),
and that (x − y)q(x, y) does not vanish on (x− , x+ ) × (y− , y+ ).
By theorem 1.3.1, each edge is either infinitely distant or has compatible normal,
and each fold is infinitely distant. The condition on the edges gives the constraints
on A(x) and B(y), using lemma 3.2.5. By an argument similar to lemma 3.2.5, each
corner is infinitely distant with respect to g0 if and only if either of the adjacent
edges is. This implies that if two edges which are not infinitely distant meet at a
corner, then that corner cannot be a fold. This gives the last condition.
Conversely, given the data described in corollary, on can construct a torus T =
R2 /Λ, and an ambitoric ansatz space A(q, A, B, T). We find that A0 : (x− , x+ ) ×
(y− , y+ ) × T ⊂ A(q, A, B, T) is a connected component of an ambitoric ansatz space
with no proper folds and each of its four edges being either infinitely distant or having
compatible normals. Then theorem 1.3.1 tells us that (A0 )gC0 is a regular ambitoric
4-orbifold which is complete with respect to g0 .
57
CHAPTER 4
Application: Compact 4-manifolds admitting Killing 2-forms
In this section, we apply our results to the setting of 4-manifolds admitting
Killing 2-forms. This will build on the work in [17], which we will summarize now.
Definition 4.0.3. A differential form ψ on a Riemannian manifold (M, g) is conformally Killing if its covariant derivative ∇ψ is of the form
∇X ψ = α ∧ X + Xβ,
∀X ∈ Γ(T M ),
for some differential forms α and β. Moreover, ψ is Killing if α = 0 and *-Killing if
β = 0.
Note that if M is oriented and ∗ is the Hodge star operator, then ψ is Killing if
and only if ∗ψ is ∗-Killing.
Proposition 4.0.2 (2.1 in [17]). Let (M, g) be a connected oriented 4-dimensional
Riemannian manifold admitting a non-parallel ∗-Killing 2-form ψ. Then on an
open dense subset M0 of M , the pair (g, ψ) gives rise to an ambiKähler structure
(g+ , J+, ω+ ), (g− , J− , ω− ), where g± = f±−2 g. Here f± :=
self-dual and anti-self dual parts of ψ.
Define vector fields K1 and K2 on such an M by
1
K1 := − α
,
2
1
K2 := (∗ψ)
(K1 ).
2
58
|ψ± |
√ ,
2
where ψ± are the
These can be used to form a rough classification:
Proposition 4.0.3 (3.3 in [17]). Any connected oriented 4-dimensional Riemannian manifold (M, g) admitting a non-parallel ∗-Killing 2-from ψ fits into one of the
following three exclusive possible cases:
1. The vector fields K1 and K2 are Killing and independent on a dense open set
U of M .
2. The vector fields K1 and K2 are Killing and K2 = cK1 for some c ∈ R\0.
3. f+ = f− on all of M , K2 = 0, and K1 is not a Killing vector field in general.
The authors go on to show that in the first case, U admits a regular ambitoric
structure (U , [g], J+ , J− ) of hyperbolic type, where g is identified with the metric that
we’ve been calling gk . In the language that we have developed, (M, U , g = gk , J+ , J− )
is a regular ambitoric manifold completion. We can then apply our classification
result to obtain:
Proof of theorem 1.3.3. From corollary 3.3.3, we find that U can be embedded as a
box-type subset of some ambitoric ansatz space. It follows that the action of 2-torus
on U extends to M with at most 4 fixed points, corresponding to the corners of
the box. We can then apply the work of [26], which classifies oriented 4-manifolds
equipped with a 2-torus action. Since the number of fixed points t is at most 4, this
classification tells us that M is (upto orientation reversal) diffeomorphic to one of
{S4 , CP2 , CP2 #CP2 , S2 × S2 , CP2 #CP2 }.
Each Hirzebruch surface is diffeomorphic to either S2 × S2 or CP2 #CP2 , so it
suffices to rule out CP2 #CP2 . To see this, we note that the convexity of μ± (U )
implies that we can ensure that the {i }ti=1 used in [26] are all equal to 1 by choosing
59
the generators of the isotropy subgroups to correspond to inward normals for the
sides of μ± (U ). This rules out the case of CP2 #CP2 , since this case has t = 4 and
1 4 = −2 3 .
Conversely, we must show that S4 , CP2 and the Hirzebruch surfaces all admit
metrics which admit non-parallel ∗-Killing 2-forms. The case of S4 with the usual
constant scalar curvature metric is explored in detail in [17]. For CP2 and the Hirzebruch surfaces, we will show that they are regular ambitoric orbifold completions of
hyperbolic type. The converse of proposition 4.0.2 [17] then provides the claimed
∗-Killing 2-form. We will use the hyperbolic normal form from section 2.2.2, so that
±
q(x, y) = x + y and the conic is given by C± = {4μ±
1 μ2 = −1}. We will take Λ to be
the standard basis for t in the coordinates given in the normal
form.⎞
⎛ ⎞ ⎛
⎛ ⎞
⎜1⎟ ⎜ 0 ⎟
⎜−1⎟
We can construct lines tangent to C± with normals ⎝ ⎠ , ⎝ ⎠ , and ⎝ ⎠,
0
−1
1
as in figure 4. Each pair of normals generates Λ, so that the moment map image is a
Delzant triangle. It is well known that the Delzant construction applied to a Delzant
triangle is CP2 .
⎛ ⎞ ⎛
⎞ ⎛ ⎞
⎜1⎟ ⎜−k − 1⎟ ⎜ 1 ⎟
We can construct lines tangent to C± with normals ⎝ ⎠ , ⎝
⎠,⎝ ⎠,
0
k
−1
⎛ ⎞
⎜−1⎟
and ⎝ ⎠ , as in figure 4. As in the triangle case, it is easy to check that the trape1
⎛ ⎞ ⎛
⎞
⎜1⎟ ⎜−k − 1⎟
zoid Δ formed by these lines is Delzant. Moreover the equation ⎝ ⎠ + ⎝
⎠=
0
k
60
t∗
CP2
Figure 4–1: The moment map image of CP2 with a hyperbolic ambitoric structure.
⎛
⎞
⎜−1⎟
k ⎝ ⎠ implies that Δ is the moment-map image of the kth Hirzebruch surface Hk
1
(see [22]). Note that to get all of the diffeomorphism types, it suffices to only construct H1 and H2 , since the odd (respectively even) order Hirzebruch surfaces are all
diffeomorphic (see for example [22]). Moreover, S2 × S2 is diffeomorphic to H2 , so it
is covered by this construction as well.
61
t∗
Hk
Figure 4–2: The moment map image of the kth Hirzebruch surface Hk with a hyperbolic ambitoric structure.
62
CHAPTER 5
Tomimatsu-Sato metrics
.
In this section we study the Tomimatsu-Sato metrics, which are a natural extension of the Kerr black-hole metrics. We show that these can be Wick rotated to
Riemannian signature in a directed generalization of the well-known Wick rotation
for Kerr. We show that these Riemannian analogues admit a regular ambitoric structure, and the more general Tomimatsu-Sato metrics admit almost-complex analogues
these structures. We then discuss the moment maps, with a complete picture in the
Kerr case, and partial results for the more general case.
The work on the Tomimatsu-Sato almost ambitoric structures is exploratory
in nature, and so far there are more questions than answers. Most troubling is
the question of which almost ambitoric structure to choose: in the integrable case,
the structure is essentially unique (given a fixed conformal class), while for the TS
metrics, we demonstrate that there is at least a two dimensional family to chose
from.
We begin in the setting of stationary axis-symmetric space-times. These are
metrics on R4 which admit two commuting Killing vector fields corresponding to
a time symmetry and a spacial rotational symmetry. In this sense we are in the
Lorentzian analogue of toric geometry. We will be studying metrics in the form of
the Papapetrou ansatz:
63
g :=
1 2γ
e (dψ 2 + dθ2 ) + ρ2 dφ2 − f (dt − ωdφ)2 ,
f
(5.1)
where f, γ and ω are functions of ψ and θ only, and ρ = sinh ψ sin θ. This metric is
Ricci-flat if and only if the following field equations hold:
0=∇·
f2
∇ω
ρ
=∇·
ρ
f 2ω
∇f +
∇ω ,
f
ρ
⎛ ⎞
(5.2)
∂
⎜ ∂ψ ⎟
where ∇ is the usual Euclidean gradient in our coordinates ⎝ ⎠ .
∂
∂θ
In [14], Ernst observes that the first field equation (5.2) is equivalent to the
existence of a function ϕ defined by
f2
∇ω = n̂ × ∇ϕ,
ρ
where n̂ is a unit vector in the azimuthal direction (i.e. n̂ ∝
(5.3)
∂
)
∂φ
and × is the
usual cross product in the variables (ψ, θ, φ). He then defines a complex potential
E := f + iϕ,
allowing the field equations (5.2) to be written as a single equation
(E)ΔE = ∇E · ∇E.
He also defines another complex potential ξ by
E=
ξ−1
,
ξ+1
64
giving an equivalent field equation
¯
(ξ ξ¯ − 1)Δξ = 2ξ(∇ξ
· ∇ξ).
If we use prolate spheroidal coordinates
x := cosh ψ,
y := cos θ,
Ernst notes that the Kerr family is represented by a very simple Ernst potential
ξ = px − iqy,
for real parameters p and q satisfying p2 + q 2 = 1. p and q can be related to mass
and angular momentum respectively by multiplication by some constants.
In [30], Tomimatsu and Sato consider solutions of the field equations corresponding to an Ernst potential ξ which is a rational function in prolate-spheroidal
coordinates. We will write ξ =
α
β
for co-prime polynomials α and β. These solutions
are indexed by a natural number δ, where
deg α = δ 2 , deg β = δ 2 − 1.
For each fixed δ, we get a family of solutions parametrized by p and q with p2 +q 2 = 1.
Tomimatsu and Sato give the explicit solutions for δ ≤ 4. In [21], Hori gives
explicit solutions for any δ ∈ N.
From the definitions of f and ξ, we can write f =
65
A
,
B
where
A =αᾱ − β β̄,
B =(α + β)(ᾱ + β̄),
5.1
Qualitative features in Lorentzian signature
In this section I will summarize some qualitative results about the TS metrics
from [30]. The Tomimatsu Sato metrics share many properties with the Kerr solutions. The hyper-surfaces at x = ±1 are event horizons. The vanishing locus of the
polynomial A divides the region outside of the event horizon into regions where
time-like and regions where
∂
∂φ
∂
∂t
is
is time-like (the ergo-sphere in the Kerr case).
The vanishing locus of the polynomial B is a singular region. Since B = (α +
β)(ᾱ + β̄), the vanishing of B is given by the vanishing of the complex polynomial
α + β. Thus the vanishing of B is along a 2 dimensional submanifold, which turns
out to be a collection of δ stationary rings within the equatorial plane (y = 0) and
outside of the event horizons (|x| > 1). In the Kerr case, the ring singularity is in the
inner region x < −1; the outer region x > 1 has no singularities. In the δ > 1 case,
there are singularities in both regions. The singularities in the outer region are called
naked singularities, since they are not hidden by an event horizon like the ones in
the inner region are. The cosmic censorship hypothesis says that there are no naked
singularities in the universe, so the δ > 1 TS metrics are un-physical. The following
diagrams show the vanishing loci of A and B in prolate spheroidal coordinates for
δ ≤ 3 and p = 3/5.
66
Figure 5–1: Qualitative picture of Tomimatsu-Sato metrics (δ ≤ 3) in Lorentzian
signature.
5.2
Wick rotation
Wick rotation is the general term to describe ways of modifying a family of
metrics in order to change the signature. This is done by allowing the coordinates
and parameters to take on complex values. Our original space bundled with its real
parameters sits as a real slice of this complexification, where the metric takes on real
values. A wick rotation is another real slice, a maximal rank sumbanifold where the
metric takes real values.
67
The easiest example, due to Wick, is that of Minkowski space, R4 with the metric
dx2 + dy 2 + dz 2 − dt2 . We can take another real slice, instead of all of the variables
being real, take the variable t to be imaginary. The metric is then the Euclidean
metric. So, the Minkowski metric is changed to the Euclidean one by rotating the
real time axis to the imaginary time axis.
It is known (see for example [18]) that this construction can be generalized to
the Kerr case by taking both time and the angular momentum (q in our notation) to
be imaginary. We show that this can be further generalized to the Tomimatsu-Sato
family:
Lemma 5.2.1. For the Tomimatsu-Sato family of solutions, q is a factor of ω, and
the functions f, γ, ωq do not depend on odd powers of q.
Proof. This is just an observation from the results in [21]. The formulas are:
f =(1 + 2u + 2w)−1
ω 2m(1 − y 2 )C
=
,
q
A
A
,
2γ = ln 2δ 2
p (x − y 2 )δ2
Where u, w, C and A are functions depending only on even powers of q, and m is
the mass constant. See [21] for the details.
Corollary 5.2.1. The region in the Tomimatsu-Sato parameter space with t and q
imaginary, −1 ≤ y ≤ 1 ≤ x is of Riemannian signature.
Proof. The lemma allows us to deduce that in this region, dt − ωdφ is imaginary,
while all of the other parts of the papapetrou ansatz (5.2) are real. The conditions on
68
x and y ensure that ψ and θ are real. The signature of the metric is then (σ, σ, σ, σ),
where σ is the sign of f .
For the remainder of the section, we will be focussed on properties of the Riemannian Tomimatsu-Sato metrics, but we will be doing computations in Lorentzian
signature. This means that when we write for example ᾱ, we are taking the complex
conjugate before Wick rotating.
5.3
Factor structure
In this section we follow [20] to describe a useful factor structure admitted by
the metric. By the definition of f in terms of the Erst potential, we see that if we
define
A =αᾱ − β β̄,
B =(α + β)(ᾱ + β̄),
then f =
A
.
B
We can also define a polynomial C in prolate spheroidal coordinates
such that ω =
C
.
A
The factor structure allows us to rewrite the metric (5.1) as
g=
1 2
B 2γ
e (dψ 2 + dθ2 ) +
λ2 (μdφ + νdt)2 − λ21 (σdφ + τ dt)2 ,
A
B
69
(5.4)
for some polynomials μ, ν, σ, τ is prolate spheroidal coordinates, and functions
λ1 , λ2 . These can be related to our previously defined polynomials by
A =λ21 τ 2 − λ22 ν 2 ,
B =μτ − νσ,
C =λ22 μν − λ21 στ,
ρ =λ1 λ2 .
ρ must factor as either
λ1 = sinh ψ, λ2 = sin θ
or
λ1 = 1, λ2 = sinh ψ sin θ.
The first option occurs for the odd δ Tomimatsu-Sato metrics, and the second option
occurs for the even δ Tomimatsu-Sato metrics. We follow [20] to define
A± := λ1 τ ± λ2 ν,
N± = λ2 μ ± λ1 σ,
so that A = A+ A− and ρB ± C = A± N∓ .
Define a complex valued derivative operator
∂± := ∂ψ ± i∂θ .
Using the field equations, one can deduce that A divides
(α∂+ β − β∂+ α)(α∂− β − β∂− α).
70
We define the factor A± of A to be the one that divides (α∂± β − β∂± α). We define
K± by
α∂± β − β∂± α = K± A± .
The Tomimatsu Sato solutions have real valued K± , but in general these could be
complex. In [20] there are more nice relations between derivatives of polynomials
like this one, but this is the only one that we need.
This computational lemma will be used later:
Lemma 5.3.1.
∂± E = −2
K± A ±
(α + β)2
Proof.
∂± E =∂±
5.4
ξ−1
α−β
β∂± α − α∂± β
K± A±
= ∂±
=2
= −2
.
2
ξ+1
α+β
(α + β)
(α + β)2
Symplectic forms
Lemma 5.4.1. If K is a Killing vector field on Ricci-flat Riemannian manifold,
then d∗ dK = 0.
Proof. From an exercise in [27], a Killing vector field K satisfies
div K = 0,
tr ∇2 K = − Ric(K).
71
But tr ∇2 = −∇∗ ∇, and the Weitzenböck formula on 1-forms is (dd∗ + d∗ d) =
∇∗ ∇ + Ric [7]. Since Ric = 0, we have
(dd∗ + d∗ d)K = ∇∗ ∇K = − tr ∇2 K = 0.
We use this to compute
d∗ dK = −dd∗ K = d div K = 0.
It follows that the projection onto the (anti-)self-dual part of dK is closed. This
gives:
Lemma 5.4.2. If K is a Killing vector field on an oriented Ricci-flat Riemannian
4-manifold, then F± (K) := dK ± ∗dK ∈ Λ± is a closed 2-form.
5.4.1
Kerr Case
In this section we see that the forms F± from the previous proposition are the
Kähler forms for a regular ambitoric structure oh hyperbolic type with conformal
class [g]. In this case, the Papapetrou ansatz can be related to the hyperbolic normal
form by the coordinate transformation
t1 = −t + 2qmφ,
t2 =
m
φ,
q
x = p cosh ψ + 1,
y = iq cos θ.
72
The ambitoric ansatz space associated with Kerr is given by
A(x + y, A(x), B(y), T), where
A(x) = x2 − 2x + q 2 ,
B(y) = −q 2 − y 2 .
A direct computation with this coordinate transformation shows that F± are transformed to ω± .
We get the following picture for A/T, where the folding hypersurfaces are drawn
in red. Note that the diagram is drawn using prolate spheroidal coordinates x =
cosh ψ and y = cos θ, not to be confused with the coordinates given by the ambitoric
ansatz.
A/T
+
y=1
-
-
Interior
y = −1
Z−
y
Exterior
-
+
x = −1
x
Z+
x=1
Figure 5–2: The domain of A/T for Kerr in prolate spheroidal coordinates. The
regions which we call interior and exterior are labelled, and have positive Riemannian
signature. The other regions with ± Riemannian signature are labelled ±. The
remaining regions have signature (2, 2).
73
The region corresponding to the exterior of the black hole has boundary edges
{x = x+ }, {x = ∞} and {y = ±α}. The edge at x = ∞ is infinitely distant, while
the other edges have compatible normals with a suitable choice of torus.
The region corresponding to x < x− corresponds to the interior of the black
hole. In Lorentzian signature, there is a ring singularity in the interior region where
the curvature diverges to infinity. This singularity transforms under Wick rotation
to the folding hypersurfaces Z± := {x ∓ y = 0} of Riemannian Kerr.
The moment map image is roughly the same for both ω± . The main component
is drawn in figure 5–3. There are two triangular domains with edges given by {x =
x− }, {y = ±α} and Z± . One of these regions is near the fold, so has the behaviour
depicted in figure 2–1, although it is hard to see in figure 5–3. The other region (not
drawn) gets mapped to a cone with edges given by {x = x− } and {y = ∓α} oriented
away from the component which was drawn.
74
t∗
y = ±α
x = x+
y = ∓α
x = x−
Figure 5–3: The momentum image of Riemannian Kerr. The conic C is drawn in
red, and {x = ∞} is drawn in blue. The exterior region is shaded with level sets of
x, while the interior region is shaded with level sets of y.
5.4.2
Almost ambitoric structures
Lemma 5.4.3. Let K be a Killing vector field for a Riemannian Tomimatsu-Sato
√
(K)
metric g. Then on the points where F± (K) is non-degenerate, J± (K) := 2 FF±±(K)
g
is an almost complex structure compatible with g with ±-orientation. Moreover
∂
.
J± (K) is integrable if and only if g is a Kerr metric and K = i ∂t
Proof. It is well known that in 4-dimensional Riemannian geometry, induces a correspondence between non-degenerate (anti-)self-dual 2-forms and metric compatible
almost complex structures. See for example [1].
75
The Weyl curvature tensor for a Tomimatsu-Sato metric is algebraically special
only in the Kerr case. Thus the only integrable complex structures are given (upto
sign) by the ambitoric structure in the Kerr case, which we’ve seen correspond to
∂
.
the Killing vector i ∂t
This motivates the following definition generalizing ambitoric structures to the
case where the almost complex structures are not integrable:
Definition 5.4.1. An almost ambitoric manifold (M, [g], ω+ , ω− , T) is an oriented
manifold M equipped with a conformal class of metrics [g], symplectic forms ω± ∈
Λ± (M ), and a torus T which has an action on M which is Hamiltonian with respect
to both ω± and isometric with respect to the almost Kähler metrics g± ∈ [g] which
are determined by the relation ω± 2g± = 2.
Proposition 5.4.1. For a Riemannian Tomimatsu-Sato metric g, with a Killing
vector field K, there is an open dense subset Mg of
M := (1 + |p|, ∞) × (−|q|, |q|) × T
with coordinates (x, y, φ, it) such that (Mg , [g], F+ (K), F− (K), T) is an almost ambitoric manifold.
Proof. By lemma 5.4.2, F± (K) ∈ Λ± are closed 2-forms. Let Mg be the subset of
M where F+ , F− and g are non-degenerate. Since g is T-invariant, so is F± (K), and
hence g± . This implies that the action of T is Hamiltonian with respect to F± (K),
since being closed and invariant is enough to ensure the existence of a moment map.
The fact that Mg is dense in M follows from the coefficient functions for g in these
√
coordinates being algebraic functions of x, y, x2 − 1, 1 − y 2 . M was chosen to be
76
the region where g has Riemannian signature corresponding to the exterior of the
black hole, although on different connected components of Mg , the signature of g
could be (+, +, +, +) or (−, −, −, −).
Note that the almost ambitoric structure depends on the choice of Killing vector
∂
was the nicest choice (since it gave
field K. Inspired by the Kerr case, where K = i ∂t
integrable complex structures), for the rest of this section we study the ambitoric
structure associated to this choice. We realize that this is a naive choice, since this
is not a geometric motivation. It would be natural to choose J+ to correspond to
one of the two principal directions of W+ (see [1]), and similarly chose J− . However,
the author has not been able to find these principal directions computationally.
5.4.3
Qualitative features in Riemannian signature
Proposition 5.4.2. On the Riemannian Tomimatsu-Sato manifolds, the 2-forms
∂
∂
∂
) = id ∂t
± ∗id ∂t
is non-degenerate away from the points where
F± := F± (i ∂t
ABK+ K− vanishes.
Proof. Define a differential operator
∂
∂
d˜ := dθ
− dψ .
∂ψ
∂θ
We can observe that (5.3) can be rewritten as
ρ ˜
dϕ,
f2
˜ = − ρ dϕ.
dω
f2
dω =
This can be used to compute F± :
77
∂
∂
± ∗id
= id(f (dt − ωdφ)) ± i ∗ d(f (dt − ωdφ))
∂t
∂t
˜ ∧ dφ ∓ f 2 dω
˜ ∧ (dt − ωdφ)
=idf ∧ (dt − ωdφ) − if dω ∧ dφ ± ρ df
F± =id
f
ρ
ρ˜
ρ˜
=idf ∧ (dt − ωdφ) + i dϕ
∧ dφ ± df
∧ dφ ∓ dϕ ∧ (dt − ωdφ)
f
f
ρ˜
± f ) ∧ dφ.
=d(if ∓ ϕ) ∧ (dt − ωdφ) + d(iϕ
f
This can be further simplified as
ρ˜
dE ∧ dφ,
f
ρ
F− =idĒ ∧ (dt − ωdφ) − d˜Ē ∧ dφ
f
F+ =idE ∧ (dt − ωdφ) +
It follows using lemma 5.3.1 that
ρ
F+ ∧ F+ =i dĒ ∧ d˜Ē ∧ dφ ∧ (dt − ωdφ)
f
ρ
=i ∂+ Ē∂− Ēdψ ∧ dθ ∧ dφ ∧ (dt − ωdφ)
f
4ρK+ K− A
dψ ∧ dθ ∧ dφ ∧ (dt − ωdφ),
=i
f (ᾱ + β̄)4
4ρK+ K− (α + β)
=i
dψ ∧ dθ ∧ dφ ∧ (dt − ωdφ).
(ᾱ + β̄)3
Similarly
F− ∧ F− = −i
4ρK+ K− (ᾱ + β̄)
dψ ∧ dθ ∧ dφ ∧ (dt − ωdφ).
(α + β)3
These forms clearly degenerate where B = (α + β)(ᾱ + β̄) vanishes.
78
It also degenerates where K± or A vanish (since ω is a rational function with a
denominator of A).
The coordinates ψ and θ are degenerate along sinh ψ = 0 and sin θ = 0 respectively, but the factor of ρ = sinh ψ sin θ ensures that this is a trivial degeneracy,
similar to the degeneracy of spherical coordinates near the poles of the 2-sphere.
In this section we investigate the qualitative features of the Wick-rotated TS
metrics, assuming that t and q are imaginary. In terms of the factor structure, for
the metric (5.4) to be Riemannian signature we must have ν and σ imaginary and
the rest of the polynomials real.
Lemma 5.4.4. In Riemannian signature, the vanishing locus of A is contained
within the vanishing locus of ρB.
Proof. A± is complex valued, with real part λ1 τ and imaginary part λ2 ν (up to
a sign). Thus A vanishing implies both λ1 τ and λ2 ν vanish. It follows that the
vanishing locus of A is contained in the vanishing locus of ρB.
Lemma 5.4.5. In Riemannian signature, all of α, ᾱ,β and β̄ are real.
Proof. This follows directly from one of the properties of (Lorentzian signature) α
and β described in [30]. The property describes that the terms in α and β that are
real have an even power of q as a factor, while the terms that are imaginary have an
odd power of q as a factor. After Wick rotating, q becomes imaginary, so all of the
terms become real.
Combining the previous two lemmas with proposition 5.4.2, we find that F± are
non-degenerate away from the curves x = ±1, y = ±1, B = 0 and K+ K− = 0. The
79
surfaces x = ±1, y = ±1 are analogous to the edges in the ambitoric case, while the
two components {α + β = 0} and {ᾱ + β̄ = 0} correspond to folding hypersurfaces.
For small δ, the vanishing locus of K+ K− is contained in the union of the other loci,
but the author has no proof of this in general.
The following diagrams show the folding hypersurfaces in prolate spheroidal
coordinates for δ ≤ 3 and p = 5/3. Note that for δ equal to 2 or 3, both folding
hypersurfaces intersect at the corners where x2 = y 2 = 1. Our intuition from the
ambitoric case is that the momentum map μ± should send one of these folding
hypersurfaces to infinity, and we shouldn’t expect convexity near the other one. In
the final section of this paper, we carefully investigate the moment map μ+ near the
corners x = 1, y = ±1 in the case of δ = 2.
80
Figure 5–4: Qualitative picture of Tomimatsu-Sato metrics (δ ≤ 3) in Riemannian
signature.
5.5
Moment maps
∂
).
In this section, we study the moment map for the symplectic form F+ = F+ (i ∂t
±
F− can be studied similarly. We can define functions (μ±
1 , μ2 ) such that F+ =
+
+
+
4
2
dμ+
1 ∧ dt + dμ2 + ∧dφ. Thus μ = (μ1 , μ2 ) : R → R is the moment map for the
81
symplectic form F+ . To compute μ+ , we use the following computation:
ρ˜
∧ dφ
F+ =idE ∧ (dt − ωdφ) + dE
f
ρ˜
dE − iωdE ∧ dφ
=idE ∧ dt +
f
˜ − iCdE ∧ dφ
=idE ∧ dt + ρB dE
A
=idE ∧ dt + ((−ρB∂θ − iC∂ψ )Edψ + (ρB∂ψ − iC∂θ )Edθ) ∧
=idE ∧ dt + ((iρB(∂+ − ∂− ) − iC(∂+ + ∂− ))Edψ ∧
+ (ρB(∂+ + ∂− ) − C(∂+ − ∂− ))Edθ ∧
dφ
A
dφ
2A
dφ
2A
=idE ∧ dt + (i(A− N+ ∂+ − A+ N− ∂− )Edψ + (A− N+ ∂+ + A+ N− ∂− )Edθ) ∧
=idE ∧ dt − (i(N+ K+ − N− K− )dψ + (N+ K+ + N− K− )dθ) ∧
+
which shows that μ+
1 = E and μ2 =
M+
,
α+β
dφ
2A
dφ
,
(α + β)2
for some complex valued polynomial M +
in prolate spheroidal coordinates satisfying
Wrx (M + , α + β) = i(N+ K+ − N− K− ),
Wry (M + , α + β) = (N+ K+ + N− K− ),
Where Wrz (F, G) = F ∂z G − G∂z F is the usual Wronskian in the variable z.
The following lemmas show that μ+ maps the level sets {x = ±1} and {y = ±1}
to lines.
Lemma 5.5.1. On the set where ρ = 0, we have
82
∂ x μ+
2
∂ x μ+
1
=
∂ y μ+
2
∂ y μ+
1
= −ω.
K± A ±
Proof. Recall that ∂± E = −2 (α+β)
2 . It follows that
dE = −
K+ A+ − K− A−
K+ A+ + K− A−
dx −
dy.
2
(α + β)
i(α + β)2
It follows that
N + K+ − N − K −
∂ x μ+
2
,
+ =
K+ A+ + K− A−
∂ x μ1
∂ y μ+
N + K+ + N − K−
2
.
+ =
K+ A+ − K− A−
∂ y μ1
If we write K± = Ke ± Ko , then recalling that A± = λ1 τ ± λ2 ν and N± = λ2 μ ± λ1 σ
gives
λ1 σKe + λ2 μK0
∂ x μ+
2
,
+ =
λ1 τ Ke + λ2 νKo
∂ x μ1
Recall that ω =
C
A
=
λ22 μν−λ21 στ
.
λ21 τ 2 −λ22 ν
∂ y μ+
λ1 σKo + λ2 μKe
2
.
+ =
λ1 τ Ko + λ2 νKe
∂ y μ1
Since ρ = λ1 λ2 , when ρ = 0, we have either λ1 = 0
or λ2 = 0. In either case, it is easy to see that
∂ x μ+
2
∂ x μ+
1
=
∂ y μ+
2
∂ y μ+
1
= −ω.
Lemma 5.5.2. The moment map μ+ sends the curves given by y = 1 and y = −1
(in other words sin θ = 0) to horizontal lines {μ+
2 = constant}.
Proof. Consider the image μ+ ({y = ±1}) viewed as a curve parametrized by x. The
tangent vector of this curve is ∂x μ+ |y=±1 . Since ρ = sinh ψ sin θ, the previous lemma
∂ μ+ = −ω|y=±1 , which vanishes since
gives that the curve’s slope is given by ∂x μ2+ x 1
y=±1
2
sin θ is a factor of the numerator of ω (see [30]). This shows that the slope of
μ+ ({y = ±1}) is 0.
Lemma 5.5.3. The moment map μ+ sends the curves given by x = 1 and x = −1
to lines, at least in the case when δ ≤ 3.
Proof. Arguing similarly to the previous lemma gives that the slope of the tangent
vector to the image μ+ ({x = ±1}) is given by −ω|x=±1 . Direct computation for
83
δ ≤ 3 yields ω|x=±1 = 2q (px + 1)|x=±1 = 2q (1 ± x), so that the slope of the curve is
constant.
These last two lemmas show that the edges of the TS metrics work as in the
Kerr case. In the following section, we will give some description of the behaviour
near the folds in the case of δ = 2.
5.5.1
δ = 2 case
Note that E =
that μ+
1 =
β
.
α+β
α−β
α+β
β
= 1 − 2 α+β
. This allows us to change a new basis for t∗ so
We do this for the rest of this section. We also find it convenient to
use variables defined as follows:
ξ = sinh ψ =
√
x2 − 1,
η = sin θ =
1 − y2.
We can then write the relevant data for the momentum map:
α =p2 x4 + q 2 y 4 − 1 − 2ipqxy(x2 − y 2 ),
= p2 (ξ 4 + 2ξ 2 ) + q 2 (η 4 − 2η 2 ) − 2ipqxy(ξ 2 + η 2 )
β =2pxξ 2 − 2iqyη 2
M + =8xy(−p2 ξ 2 + q 2 η 2 ) + 4iqx(4ξ 2 − η 4 + 2η 2 ) +
4y 2 2
(4q η − 2p2 ξ 2 − p2 ξ 4 )
p
+ 4ipq(2ξ 2 + 2η 2 + ξ 4 − η 4 )
Taking the limit as x → ∞, we find μ+
2 =
M+
α+β
is bounded, while μ+
1 =
β
α+β
This means that the edge {x = ∞} gets mapped by μ+ to the line {μ+
1 = 0}.
84
→ 0.
Restricting to y = ±1, we get
α =ξ 2 (p2 (ξ 2 + 2) − 2ipqxy),
β =2pxξ 2
M + = p4 (iq − y)(α + β),
4
so that μ+
2 |y=±1 = p (iq − y). Taking the limit as x → 1 along y = ±1 gives
lim
x→1
μ+
1 |y=±1
β
2pxξ 2
|y=±1 = lim 2 2 2
= lim
x→1 α + β
x→1 ξ (p (ξ + 2) − 2ipqxy) + 2pxξ 2
1
.
=
p − iqy + 1
The line containing {x = 1} is then the line connecting the points
1
4
1
4
, (iq − 1) ,
, (iq + 1) .
p − iq + 1 p
p + iq + 1 p
At these points, α, β and M + vanish. To study these points, we study how the
momentum coordinates vary near the points. Since both ξ and η vanish at these
points, we can think of them as infinitesimals locally. Let’s first consider the first
point (y = x = 1). We compute
iq
(p + 1 + iq)η 6 + O(η 8 ) + O(η 2 ξ 2 ),
p
q2
1
(α + β) =
η 6 + O(η 8 ) + O(η 4 ξ 2 ) + O(η 2 ξ 4 ) + O(ξ 6 ).
β−
p − iqy + 1
4(p + 1 − iq)
M + − p4 (iq − 1)(α + β) =
The ratios between these quantities give the slopes in momenta. We approach ξ =
η = 0 along a curve ξ = κη r .
85
For r < 12 , the lowest order term is ξ 6 , which does not occur in the first quantity.
This gives a slope of 0, which corresponds to the horizontal line {y = 1}.
For
1
2
< r < 2, the lowest order term is ξ 2 η 2 , which only occurs in the second
quantity. This corresponds to a vertical line.
For r > 2, the lowest order term is η 6 . One can compute that the corresponding
, which coincides with the slope for the line {x = 1}.
slope is − 8(p+1)
ipq
For r ∈ { 12 , 2}, varying κ varies the slope, interpolating between the slopes for
r slightly smaller or slightly larger.
In summary, the tangent cones at x = 1, y = ±1 can be described as follows.
One boundary of the cone is the horizontal ray corresponding to y = ±1. The other
boundary is the ray corresponding to the line x = 1. As r passes through the value
1, the slope passes through a vertical slope, reversing the direction of sweeping as it
passes. This results in the following diagram:
86
Figure 5–5: Moment map image of TS2. On the left, the domain which we consider
is shaded in, using prolate spheroidal coordinates. On the right the boundary of the
image is presented, where the image lies on the right side of the boundary.
The horizontal lines correspond to y = ±1. The left-most vertical line corresponds to x = ∞. The right-most vertical line corresponds to (x, y) = (1, −1). The
sloped line corresponds to x = 1, and is shown dashed in the top region since it is not
a boundary of the image that we are highlighting. This dotted ray and the adjacent
vertical line form the image of the region {x > 1, y 2 < 1, α + β < 0}. The region
{x > 1, y 2 < 1, ᾱ + β̄ < 0} gets mapped to the convex hull of a curve C which has
some representative points drawn as dots. The following diagram shows these dots
more clearly:
87
Figure 5–6: Moment map image near the fold of TS2.
We find that the momentum map acting on {x > 1, y 2 < 1} is not injective:
The cone generated by the dashed ray and the adjacent vertical ray is mapped to
two twice, once by each of the connected components of the domain adjacent to
{α + β = 0}. The convex hull of C is also mapped to more than once. The line
μ+
2 =
4
(iq
p
− 1) (corresponding to y = 1) divides the convex hull of C into two
regions. The bottom region is mapped to twice, one for each connected component
of the domain adjacent to {ᾱ + β̄ = 0}. The upper region is mapped to three times,
so that even the main component (shaded in figure 5.5.1) is not injectively mapped
by the moment map. This is behaviour that does not occur for the moment maps
on ambitoric ansatz spaces, which we saw were injective on connected components.
88
CHAPTER 6
Appendix: Busemann completions
The Busemann completion of a Riemannian manifold is a set which includes the
Cauchy completion as well as elements corresponding to directions at infinity. Our
goal is to relate the Busemann completions of different metrics on the same manifold.
We follow the work of [16], although we make some cosmetic changes of definitions
to suit our goals.
Definition 6.0.1. Let (M, g) be a Riemannian manifold. Let
C(M ) := c : [0, ∞) → M piece-wise smooth curve / ∼,
where ∼ is equivalence of curves by reparametrization. We will refer to a class of
curves [c] ∈ C(M ) simply by c, thinking of it as an unparametrized curve. For
c ∈ C(M ), we define the Busemann function
bgc
: M → R ∪ {∞} : x → lim
s→∞
s
0
ċ(t)g dt − dg x, c(s)
,
where dg is the distance with respect to g. bc is well defined, since it is clearly
invariant under reparametrization of c.
We reproduce some facts from [16]:
Proposition 6.0.1 (Proposition 4.15 in [16]).
• If bgc (x) = ∞ for some x ∈ M , then bgc ≡ ∞.
89
• If c ∈ C(M ) has finite length, then there exists some x̄ in the Cauchy completion
MCg of M such that
bgc (x) = lengthg (c) − dg (x, x̄).
(6.1)
Conversely, for every x̄ ∈ MCg , there exists a c ∈ C(M ) with finite length
satisfying (6.1).
Definition 6.0.2. Let B(M )g := {bc }c∈C(M ) \{∞} be the set of finite Busemann
functions. The Busemann completion of (M, g) is given by MBg := B(M )g /R, where
R acts by the addition of constant functions.
Note that the second part of proposition 6.0.1, we can identify the Cauchy
completion MCg as a subset of the Busemann completion MBg via the map x̄ →
[−dg (x̄, ·)].
We define an equivalence relation on C(M ) by
c1 ∼g c2 ⇐⇒ ∃r ∈ R : bgc2 = bgc1 + r.
The map [c] → [bc ] gives us an isomorphism
g
C(M )/∼g ∼
= MB ∪ {[c ∈ C(M ) : bgc ≡ ∞]}.
This allows us to interpret MBg as a set of equivalence classes of curves. This will
help us understand how MB depends on the metric since the curves are independent
of the metric; only the equivalence relation changes.
Let g1 and g2 be two Riemannian metrics on M . The rest of the appendix will
be dedicated to proving the following proposition:
90
Proposition 6.0.2. If MCg2 is compact, then every point x̄ ∈ MCg1 can be represented
by a point x̄ ∈ MCg2 , in the sense that there is a curve c ∈ C(M ) such that [c]g1 = x̄
and [c]g2 = x̄ .
The following lemma will do most of the work for us:
g1
Lemma 6.0.4. Let (xn )∞
n=1 be a sequence in M which g1 -converges to some x̄ ∈ MC .
Then there exists a g2 -geodesic curve c such that x̄ lies in the g1 -closure of the image
of c.
Proof. Let p ∈ M . For each n ∈ N, let cn ∈ C(M ) be a g2 -geodesic ray starting from
x0 and passing through xn . Each cn is generated by the exponential mapping from
a direction un ∈ S(Tp M ) in the unit sphere bundle at p. Since S(Tp M ) is compact,
∞
there exists a subsequence (unk )∞
k=1 of (un )n=1 converging to some u∞ . Let c be the
geodesic generated by u∞ .
Let > 0. Since limk→∞ unk = u∞ , there exists a k0 ∈ N such that unk ∈ B (u∞ )
for all k > k0 , where B (u∞ ) is the -ball around u∞ in S(Tp M ). Consider the set
S := {expgp2 (tu) : t ∈ [0, ∞), u ∈ B (N∞ ), tu ∈ domain(expgp2 )} ⊆ M.
For each k > k0 ,
S ⊃ Image(cnk ) # xnk .
Thus
S ⊃ {xnk }k>k0 # x̄,
where · indicates the closure in MCg1 . Since > 0 was arbitrarily chosen, we find that
x̄ ∈
S̄ = Image(c).
>0
91
Lemma 6.0.5. Let g3 be a third metric on M . Let (xn )∞
n=1 be a sequence in M
which g1 -converges to some x̄1 ∈ MCg1 and g3 -converges to some x̄3 ∈ MCg3 . Then
there exists a g2 -geodesic curve c such that x̄j lies in the gj -closure of the image of c
for each j ∈ {1, 3}.
Proof. This follows from applying the previous lemma twice (once using g3 in place
of g1 ), and noting from the proof that resulting curve would be the same, since it
doesn’t depend on g1 .
We would like to build an equivalence relation which includes the information
of both ∼g1 and ∼g2 . We define the relation ∼gg12 by
c1 ∼gg12 c2 ⇐⇒ c1 ∼g1 c2 or c1 ∼g2 c2 .
Note that ∼gg12 is not an equivalence relation in general, since it may not be transitive.
To see this, consider the following example. Let M = S1 ×S1 ×R. Let g1 (respectively
g2 ) be a metric which contracts the first (respectively second) S1 factor to a point at
the origin of R. For example, using coordinates (θ1 , θ2 , r) ∈ S1 × S1 × R,
g1 = r2 dθ12 + dθ22 + dr2 ,
g2 = dθ12 + r2 dθ22 + dr2 .
92
Let {t1 , t2 , t3 } be different points in S1 . Consider the constant curves on M :
c1 = (t1 , t2 , 0),
c2 = (t3 , t2 , 0),
c3 = (t3 , t1 , 0).
We see that c1 ∼g1 c2 ∼g2 c3 , but c1 ∼gg12 c3 .
Consider the equivalence relation ≈ on C(M ) defined by c1 ≈ c2 if and only if
there exists a finite set of curves {Ci }m
i=1 ⊂ C(M ) such that
c1 ∼gg12 C1 ∼gg12 C2 ∼gg12 · · · ∼gg12 Cm ∼gg12 c2 .
Define MB≈ := (C(M )/ ≈)\{[c ∈ C(M ) : bgc 1 = gcg2 ≡ ∞]}. Let πi : MBgi → MB≈ be
the natural quotient map for each i ∈ {1, 2}.
Lemma 6.0.6. π1 (MCg1 ) ⊆ π2 (MBg2 ).
Proof. Assume that there exists x̄ ∈ MCg1 such that π1 (x̄) ∈
/ π2 (MBg2 ). Unravelling the
definitions, this means that for each curve c ∈ C(M ) g1 -converging to x̄, bgc 2 ≡ ∞.
Fix c(t) a parametrized curve g1 -converging to x̄. Since bgc 2 ≡ ∞, c has infinite
g2 -length by proposition 6.0.1.
Claim: The image of c is unbounded with respect to g2 .
Proof of claim. Assume that the claim is false. Then the image of c lies in some
closed set B ⊂ MCg2 which is bounded with respect to g2 . Thus there exists a
∞
sequence (tn )∞
n=1 in R such that c(tn ) n=1 converges in g2 to some x̄ ∈ B. Applying
lemma 6.0.5 we find a g2 geodesic curve c which g1 -converges to x̄ and g2 converges
93
to x̄ . Since c g2 -converges to x̄ it has finite g2 -length. Thus bgc2 ≡ ∞, contradicting
the assumption on x̄ since c g1 -converges to x̄.
The claim allows us to find a sequence (tn )∞
n=1 in R such that for any x0 ∈
M , limn→∞ dg2 x0 , c(tn ) = ∞. Applying lemma 6.0.4 to this sequence gives a g2 geodesic curve c which g1 -converges to x̄. From [29] the function
t : S(Tx0 M ) → (0, ∞] : u → sup t > 0 : dg2 (x0 , expx0 tu) = t
is continuous. Thus
t(u∞ ) = lim t(unk ) ≥ lim dg2 x0 , c(tnk ) = ∞,
k→∞
k→∞
so that t(u∞ ) = ∞, where (unk )∞
k=1 is the sequence constructed in lemma 6.0.4. In
other words, the geodesic c generated by u∞ is g2 -distance minimizing. Thus
bgc2 (x0 )
= lim
s→∞
s
0
ċ∞ (t)g2 dt − dg2 (x0 , c∞ (s))
= lim 0 = 0,
s→∞
so that bgc2 ≡ ∞. This contradicts the assumption on x̄, since c g1 -converges to x̄.
Lemma 6.0.7. If MCg is compact, then MCg = MBg .
Proof. Let x0 ∈ M . Since MCg is compact, the diameter of M is bounded, so that
the function dg (x0 , ·) is bounded. This implies that bgc (x0 ) = ∞ for all infinite length
curves c. The result then follows from proposition 6.0.1.
Combining the previous two lemmas produces a proof of proposition 6.0.2.
94
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