Geometric Flows and Related Topics
Workshop at the School of Mathematical Sciences
Queen Mary University of London
Titles and Abstracts
Andre Neves (Imperial)
Index estimates for Min-max Theory
Ben Sharp (Pisa)
Compactness theorems for smooth minimal hypersurfaces with bounded index and area
We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed
Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of
closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is
quantised in terms of the total curvature of some limit surface, plus a sum of total curvatures of
complete properly embedded minimal hypersurfaces in Euclidean space, yielding qualitative control
on the topology of the hypersurfaces. This is a joint work with Reto Buzano.
Giuseppe Tinaglia (King’s)
The geometry of constant mean curvature disks embedded in R3
In this talk I will survey several results on the geometry of constant mean curvature surfaces embedded in R3 . Among other things I will prove radius and curvature estimates for nonzero constant
mean curvature embedded disks. It then follows from the radius estimate that the only complete,
simply connected surface embedded in R3 with constant mean curvature is the round sphere. This
is joint work with Bill Meeks.
Peter Topping (Warwick)
Sharp local decay estimates for the Ricci flow on surfaces
There are many tools available when studying 2D Ricci flow, equivalently the logarithmic fast diffusion equation, but one has always been missing: how do you get uniform smoothing estimates in
terms of local L1 data, i.e. in terms of local bounds on the area. The problem is that the direct
analogue of the geometrically less-useful Lp smoothing estimates for p > 1 are simply false. In
this talk I will explain this problem in more detail, and show how to get around it with a new local
decay estimate. I also plan to sketch the proof and/or give some applications. Joint work with
Hao Yin.
Melanie Rupflin (Oxford)
Harmonic-Ricci flow on surfaces
While Ricci flow evolves a metric g only by conformal changes if the domain M is a surface, this
property is not shared by the so called Harmonic-Ricci-flow which is obtained by coupling Ricci
flow with the harmonic map heat flow of maps u : M → N , N a fixed Riemannian manifold. As
we shall discuss in this talk we can however split up the evolution of the metric component of the
flow into three different types of movements, conformal changes, pull-backs by diffeomorphisms
and horizontal changes, each of which we can then analyse by different methods. As a result we
obtain (for large enough coupling constants, domains of positive genus and general targets N )
that solutions of the renormalised harmonic Ricci-flow remain smooth for all times. The presented
results are joint work with Reto Buzano.
Pau Figueras (Queen Mary)
The stationary Einstein equations and non-existence of stationary Ricci solitons
Previously the deTurck ’trick’ has been used to render the stationary Einstein equations a well posed
elliptic system that may be solved numerically by geometric flow or directly. Whilst in the static
case for pure gravity with zero or negative cosmological constant there is a simple proof that solving
the modified ’harmonic’ Einstein equations leads to a solution of the original Einstein system – i.e.
not a Ricci soliton – in the stationary case this argument no longer works. Here we provide a new
argument that the static result may be extended to the case of stationary spacetimes. In particular
we prove the non-existence of steady and expanding stationary Lorentzian Ricci solitons, in the
case that the vector field is othogonal to the timelike Killing vector. Our argument works for a
globally timelike Killing vector, but also allows Killing horizons provided the spacetime moves rigidly.
Cecile Huneau (Cambridge)
Stability of Minkowski space-time with a translation Killing field
In the presence of such a translation symmetry, the 3 + 1 vacuum Einstein equations reduce to
the 2 + 1 Einstein equations with a scalar field. We work in generalised wave coordinates. In this
gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The
main difficulty in this paper is due to the weak decay of free solutions to the wave equation in 2
dimensions compared to 3 dimensions. This weak decay seems to be a deterrent for proving a stability result in the usual wave coordinates. In this talk we present a suitable generalized wave gauge
in which our system has a “cubic weak null structure”, which allows for the proof of global existence.
Kai Zheng (Warwick)
Remark on the Lagrangian mean curvature flow
We consider the asymptotic behaviour of global solution to the Lagrangian mean curvature flow.
Dejan Gajic (Cambridge)
Sharp decay estimates for waves on black holes and Price’s law
We discuss sharp decay rates for the wave equation on black hole backgrounds (known as "late-time
tails" in the physics literature) and present a new physical space method for obtaining the sharp
decay rates on Schwarzschild black holes that are given by Price’s law. This work has been done
jointly with Yannis Angelopoulos and Stefanos Aretakis.
Dan Ketover (Imperial)
Sharp entropy bounds and min-max theory
I will discuss how min-max theory can be applied to address a problem in mean curvature flow that
can be considered the parabolic analog to the Willmore conjecture. This is joint work with X. Zhou.
Otis Chodosh (Cambridge)
The index and topology of minimal surfaces in R3
I’ll discuss the relationship between the topology and index of minimal surfaces in R3 . This is joint
work with Davi Maximo.
Hassan Jolany (Lille)
Song-Tian program
Existence of canonical metric on a projective variety was a long standing conjecture which the
major part of this conjecture is about varieties which do not have definite first Chern class (most
of the manifolds do not have definite first Chern class). There is a program which is known as
Song-Tian program for finding canonical metric on canonical model of a projective variety by using
minimal model program. In this talk, we investigate conical Kähler-Ricci flow on holomorphic fiber
spaces π : (X, D) → B whose generic fibers are log Calabi-Yau pairs, c1 (KB + D) < 0, and D
is a simple normal crossing divisor on X (we consider the case c1 (KB ) = 0, c1 (KB ) < 0 and
c1 (KB ) > 0). We show that there is a unique conical Kähler-Einstein metric on (X, D) which is
twisted by logarithmic Weil-Petersson metric plus current of integration by introducing Log YauVafa semi Ricci flat metrics which is semi positive metric with pole singularities. We introduce
a logarithmic canonical measure and show that its inverse is analytic Zariski decomposition. For
positive case we show some partial results by additional condition on logarithmic CM -line bundle.
Moreover, we show that along the conical Kähler-Ricci flow the metric collapsing of (X \ D, ω(t))
converges exponentially fast in C 0 -topology (in metric sense) hence in Gromov-Hausdorff topology
to ωB .
Tobias Lamm (Karlsruhe)
Conformal Willmore Tori
In this talk I am going to present recent existence and non-existence results for conformal Willmore
Tori in R4 which were obtained in a collaboration with Reiner M. Schätzle (Tübingen).
Huy Nguyen (Queensland)
The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds
In joint with Reto Buzano, we generalise the classical Chern-Gauss-Bonnet formula to a class of
4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends
a result of Chen-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature
and positive scalar curvature at the ends and at the singularities, we obtain a new Chern-GaussBonnet formula with error terms that can be expressed as isoperimetric deficits.
Robert Haslhofer (Toronto)
Mean convex level set flow in general ambient manifolds
We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature,
and the ratio between the norm of the second fundamental form and the mean curvature. In
particular, the estimates remove a stumbling block that has been left after the work of White
and myself and Kleiner, and thus allow us to extend the structure theory for mean convex level
set flow to general ambient manifolds of arbitrary dimension. This is joint work with Or Hershkovits.
Esther Cabezas-Rivas (Frankfurt)
Almost-“whatever” curvature conditions
During the talk we will review the topological implications of almost flat (AF) and almost nonnegatively curved (ANNC) manifolds. The study of the structure of limit spaces coming from
sequences of manifolds satisfying suitable curvature conditions led us to a generalization of the
celebrated AF manifold theorem by Gromov and to prove the vanishing of the Â-genus for ANNC
manifolds. The use of Ricci flow techniques allows us to obtain the corresponding improvement
of the already improved version by Ruh of Gromov’s theorem. Finally we will mention manifolds
with almost non-negative curvature operator (ANCO), whose classification is an intriguing open
problem. By exploiting new heat kernel estimates for the Ricci flow, we will obtain that, roughly
speaking, noncollapsed ANCO manifolds also admit metrics with nonnegative curvature operator.
Panagiotis Gianniotis (UCL)
The size of the singular set of a Type I Ricci flow
In this talk, I will consider a decomposition of the singular set of a Type I Ricci flow according
to the number of Euclidean factors split by the tangent flows. It turns out that each set of this
decomposition satisfies certain volume decay estimates, as the flow approaches the singular time.
Such estimates may be seen as analogues for the Ricci flow of classical Hausdorff dimension estimates that are valid in other contexts, such as energy minimizing maps.
Jason Lotay (UCL)
The Laplacian flow in G2 geometry
A key challenge in Riemannian geometry is to find Ricci-flat metrics on compact manifolds, which
has led to fundamental breakthroughs, particularly using geometric analysis methods. All non-trivial
examples of such metrics have special holonomy, and the only special holonomy metrics which can
occur in odd dimensions must be in dimension 7 and have holonomy G2 . I will describe recent
progress on a proposed geometric flow method for finding metrics with holonomy G2 , called the
Laplacian flow. This is joint work with Yong Wei.
Mario Micallef (Warwick)
Minimal surfaces of high codimension: isotropicity, holomorphicity and stability
The notions mentioned in the title are roughly related as follows. Isotropicity of a minimal surface
is characterised by the vanishing of certain holomorphic differentials. Holomorphic curves in a
complex torus with a flat metric are precisely the minimal surfaces which are maximally isotropic.
And it is well known that a surface which is holomorphic in a Kähler manifold minimizes area in
its homology class. I will present various results in this area and mention some open problems.
In particular, I will discuss the deformation of a holomorphic curve in a complex torus with a flat
metric to a minimal surface which is sufficiently highly (but less than maximal!) isotropic. The
holomorphicity of stable minimal surfaces which are isotropic to the same degree will also be described. This is joint work with Elisabeta Nedita and it is related to (some old) work with Claudio
Arezzo and Jon Wolfson.
Sylvain Maillot (Montpellier)
Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature
A few years ago F. C. Marques proved that if M is a compact, orientable 3-manifold which admits
a Riemannian metric of positive scalar curvature, then the moduli space of such metrics (modulo
diffeos of M ) is path-connected with respect to the smooth topology. We present a generalization
of this theorem to possibly non-compact manifolds, working with the space of complete metrics
with bounded geometry and scalar curvature >1. The proof combines a version of Ricci flow with
surgery and topological methods. (joint with Laurent Bessières, Gérard Besson and Fernando Codá
Marques)