## Detailed version

### Spring Term 2016

Marco Radeschi. Metrics on spheres with all geodesics closed. Thursday 21st January, 3-4pm, Huxley 342
Abstract: Riemannian manifolds with all geodesics closed have been studied since the beginning of last century, when Zoll showed the existence of a non-round 2-dimensional sphere of this type. Among the many open problems on the subject, a conjecture of Berger states that for any simply connected manifold with all geodesics closed, the geodesics must have the same length. In this talk, I will show recent joint work with B. Wilking, where we use equivariant Morse theory on the free loop space to prove that the Berger’s conjecture holds for every sphere of dimension >3.

Ivaldo NunesMinimal surfaces with free boundary. Thursday 4th February, 3-4pm, Huxley 342
Abstract: In this talk we will discuss the free boundary problem for minimal surfaces. Our main goal is to deal with the existence of such surfaces. In particular, we prove that every strictly convex compact domain of  the Euclidean space R^3 contains a properly embedded free boundary minimal surface which is topologically equivalent to an annulus. This is a joint work with D. Máximo (Stanford University) and G. Smith (Universidade Federal do Rio de Janeiro – UFRJ).

Gregory Chambers. The Log-Convex Density Conjecture. Thursday 11th February, 3-4pm, Huxley 342
Abstract: In this talk, I will explain the main components of the proof of the Log-Convex Density Conjecture. This conjecture, due to K. Brakke, asserts that balls centered at the origin are isoperimetric regions in Euclidean space endowed with a positive density which is smooth, radially symmetric, and log-convex. I will also show that these are the only isoperimetric regions, unless the density is constant on some ball.

These methods have recently been used to solve a similar problem; in Euclidean space with density $f(x) = |x|^p$, $p > 0$, balls whose boundaries pass through the origin are isoperimetric regions.  I will explain how the components of the proof of the first theorem can be adapted to prove this one as well.

Anton Petrunin. On the total curvature of minimizing geodesics on convex surfaces. Thursday 18th February, 3-4pm, Huxley 342
Abstract: We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space. This is a joint work with Nina Lebedeva.

Miguel Ortega. nabla-Einstein 3-Sasakian homogeneous manifolds Wednesday 24th February, 3:30-4:30pm, Huxley 139
Abstract: A Riemann-Cartan manifold is a Riemannian manifold with a compatible connection, namely paralelizing the metric. For homogeneous Riemannian manifolds admitting a reductive decomposition, K. Nomizu showed that the set of such connections is bijective to a finite dimensional linear space of bilinear maps. In addition, we say that the Riemann-Cartan manifold has skew-torsion when it is possible to construct a skew-symmetric 3-form by using the torsion of the connection. Among Riemann-Cartan manifolds with skew-torsion, I. Agricola and A.C. Ferreira considered nabla-Einstein manifolds, i.e., when the symmetric part of the Ricci tensor is a multiple of the metric. Next, C. Draper, A. Garvin, F. J. Palomo studied spheres which are symmetric spaces and nabla-Einstein. Now, we move to a list of 3-Sasakian homogeneous manifolds. In all these cases, the dimension of the space of invariant, compatible, skew-symmetric torsions is always 10. Next, by using the natural almost 3-contact structures in such manifolds, we compute the set of invariant, compatible, skew-symmetric connections such that our manifolds become nabla-Einstein. We show that when the dimension is bigger than 7, only the Levi-Civita connection can exist. Finally, we also adapt the concept of pseudo-Einstein manifolds from real hypersurfaces in quaternionic space forms, to our setting.

Simon Donaldson. G2 manifolds, Torelli-type questions and maximal submanifolds. Thursday 25th February, 3-4pm, Huxley 342
Abstract:  We will recall some general background in the theory of 7-manifolds with holonomy G2 and Calabi-Yau 3-folds. In particular we will explain a variational point of view due to Hitchin. We will go on to discuss an “adiabatic limit” of these questions, and explain that it is a global form of the “maximal submanifold” equation,  for submanifolds in a space of indefinite signature.  We will recall some helpful features of this theory, involving the sign of the Ricci curvature. If time allows we will discuss a  possible strategy for attacking uniqueness questions of  “Torelli type”.

Luca Spolaor. Regularity theory for two dimensional almost area minimizing currents. Thursday 10th March, 3-4pm, Huxley 342
Abstract: Building upon the Almgren’s big regularity paper, Chang proved in the eighties that the singularities of area-minimizing integral 2-dimensional currents are isolated. His proof relies on a suitable improvement of Almgren’s center manifold together with an asymptotic analysis of the frequency function. In recent joint works with Camillo De Lellis and Emanuele Spadaro we give a complete proof of the existence of the center manifold, only sketched by Chang, and extend his theorem to two classes of currents which are “almost area minimizing”, namely spherical cross sections of area-minimizing 3-dimensional cones and semicalibrated currents.

### Autumn Term 2015

Spyros Alexakis. The impossibility of periodic motion in general relativity Monday 12th October, 2-3pm, Huxley 341
Abstract: The problem of motion of bodies in general relativity dates back to the early days of the theory. Initially considered in the slow-motion approximation, the derivation of the equations of motion to first post-Newtonian order is due to Einstein, Infeld and Hoffman, with much more precise approximations obtained since. A natural question considered in this connection is whether there exist solutions which are periodic in time–the problem of eternal return. For asymptotically simple space-times, we show that this is not possible, at least near infinity. We relate this to the problem of reconstructing a solution of the Einstein equations from knowledge of the radiation it has emitted towards infinity. Joint work with V. Schlue, and (partly) A. Shao.

Niels Moller. The gluing perspective on non-compactness of moduli spaces of minimal surfaces Monday 19th October, 2-3pm, Huxley 341
Abstract: It is expected (Hoffman-Meeks + Ros) that M(g,r), the space of final total curvature complete embedded minimal 2-surfaces in R^3 of genus g and r ends, is empty when g + 2 < r and for r ≥ 4 is non-compact whenever non-empty. Families exhibiting such non-compactness come about by considering the gluing of coaxial catenoids (as Kapouleas 1997), but uniformly allowing intersection angles to degenerate to zero. I will discuss the most symmetric case, which viewed at the right scale is really a doubling of the flat plane. The resulting surfaces turn out to only be Alexandrov embedded. That this must be so follows already from a uniqueness theorem of Ros from 2006, but I will also explain how it is seen directly in the gluing procedure where one can track the influence from shearing on the final result. This work is joint with Stephen Kleene.

Otis Chodosh. Minimal surfaces with bounded index in three-manifolds Monday 2nd November, 2-3pm, Huxley 341
Abstract: I will discuss the local picture of a degenerating sequence of embedded minimal surfaces with uniformly bounded index in a three manifold. This leads to a “controlled surgery” result, which allows us to prove a variety of compactness results for such surfaces. This is joint work with Dan Ketover and Davi Maximo.

Toby Wiseman. Applications of geometry to quantum field theory using the AdS-CFT correspondence Moday 16th November, 2-3pm, Huxley 341
Abstract: The aim of the talk will be to review in a pedagogical way how the ‘AdS-CFT correspondence’, which derived originally in string theory, turns certain questions in quantum field theory into geometric problems. Of most probably interest to the audience are a subclass of these problems which are related to the geometry of asymptotically hyperbolic Einstein spaces. I will then discuss a specific example which nicely illustrates how simple geometric analysis provides a very interesting result for certain quantum field theories. On the physics side this results relates to the behaviour of the `Casimir energy’ of the field theory. From the geometric perspective the result is related to a necessary condition for existence of asymptotically hyperbolic Einstein spaces with a specific isometry.

Marco Guaraco. Min-max, phase transitions and minimal hypersurfaces Monday 30th November, 2-3pm, Huxley 341
Abstract: There is a strong correspondence between critical points of in the theory of phase transitions and critical points of the area functional in theory of minimal hypersurfaces. Historically, this connection have been well established the case of minima or stable critical points. We use ideas from Pitts to extend these results to the case of unstable critical points of any index. As an application we obtain a new min-max method for constructing embedded minimal hypersurfaces in an arbitrary closed manifold of any dimension. Our approach is variational, but it is substantially different from Almgren-Pitts theory. We also study the correspondence of critical points from a “global” variational point of view in the case of those obtained by min-max methods. In particular, we compare our construction of a minimal hypersurface with that of Almgren-Pitts and its refinement by Simon-Smith in dimension 3.

Marc Lackenby. Non-positively curved surfaces and the topology of knots Monday 7th December, 2-3pm, Huxley 341
Abstract: One of the useful consequences of the geometrisation of 3-manifolds is that it leads to control over embedded surfaces. If one makes an essential surface minimal within a hyperbolic 3-manifold, the surface then inherits a negatively curved metric, and this has geometric and topological applications. In my talk, I will explain how in the case of knot complements, one can arrange for embedded essential surfaces to inherit a non-positively curved metric, even when no hyperbolic metric on the knot complement is given. This has some striking consequences, particularly for the knot classification problem.

### Summer Term 2015

Yakov Shlapentokh-Rothman. Stability and Instability of Scalar Fields on Kerr Spacetimes Thursday 23rd April, 2-3pm, Huxley 139
Abstract: I will discuss some stability and instability results for wave and Klein-Gordon equations on sub-extremal Kerr exterior backgrounds. More specifically, for the wave equation we will show that general finite energy solutions have a uniformly bounded energy and satisfy an integrated local energy decay estimate. In contrast, for the Klein-Gordon equation we will construct finite energy solutions which grow exponentially. A primary goal of the talk will be to explicitly describe the close connection between these analytical results and various well-known and less well-known geometric features of the Kerr spacetime.

Nikolai Nowaczyk . Existence of Dirac Eigenvalues of higher Multiplicity Thursday 30th April, 3-4pm, Huxley 140
Abstract: We prove that on any compact spin manifold of dimension m congruent 0,6,7 mod 8, there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by catching the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.

Martin Taylor. Stability of Minkowski Space for the Massless Einstein–Vlasov system Thursday 7th May, 2-3pm, Huxley 140
Abstract: Massless collisionless matter is described in general relativity by the massless Einstein–Vlasov system. Given asymptotically flat Cauchy data for this system which is sufficiently close, in a suitable sense, to the trivial solution, Minkowski space, the resulting maximal development exists globally in time and asymptotically decays appropriately. This can be shown via a reduction to the corresponding result for the vacuum Einstein equations, a monumental result first obtained by Christodoulou–Klainerman in the early ’90s. A key step in the proof is to estimate certain Jacobi fields on the mass shell, a submanifold of the tangent bundle of the spacetime endowed with the Sasaki metric.

Yong Wei. On the Laplacian flow for closed G_2 structures. Thursday 14th May, 2-3pm, Huxley 140
Abstract: The Laplacian flow for closed G_2 structure was introduced by R. Bryant in 1992, which provides a potential tool for studying the existence of torsion-free G2 structures, and thus Ricci-flat metrics with exceptional holonomy G2, on a 7-dimensional manifold. This flow has the short-time existence result and then motivates the study of the long time behavior of the flow. The main goal is to find some condition on the initial closed G_2 structure such that the Laplacian flow will exist for all time and converge to a torsion free G_2 structure.

In this talk, I will start with the background of G_2 structure and the motivation of introducing the Laplacian flow; then I will present my recent work (joint with Jason D. Lotay) on the foundational theory of the Laplacian flow, including the Shi-type derivative estimate, uniqueness, long-time existence obstruction and compactness; finally, I will show the stability of torsion-free G_2 structure and some results on the compact soliton solution.

Carlo Sinestrari. Mean curvature flow of submanifolds of complex projective space Thursday 28th May, 2-3pm, Huxley 140

Abstract: We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and if it satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviours: either the submanifold shrinks to a round point in finite time, or it converges smoothly to a totally geodesic limit in infinite time.
A similar alternative was already known to hold for the mean curvature flow of submanifolds on the sphere from previous work by Huisken and Baker. The results are in collaboration with G. Pipoli (Grenoble).

Christos Mantoulidis. Bartnik quasilocal mass of apparent horizons Thursday 18th June, 2-3pm, Huxley 140
Abstract: In the context of 3-manifolds that arise as time symmetric initial data sets for Einstein’s field equations that are Euclidean near infinity, non-negative scalar curvature and the presence of compact minimal surfaces restricts (from above) the speed of decay to flatness. Bartnik’s quasilocal mass for compact minimal surfaces can be used to measure the optimal asymptotic decay to flatness in the presence of said surface. By the Riemannian Penrose inequality this can be bounded on one side in terms of the area of the surface. In joint work with Rick Schoen we have shown this one-sided bound on Bartnik’s mass is, generically, an exact equality. Furthermore, in the spirit of Thorne’s hoop conjecture we find that Bartnik’s mass cannot be suitably bounded in terms of the length of the shortest or the min-max geodesic like we know it is in terms of area.

Rui Albuquerque. Higher degree Cartan structural equations Thursday 25th June, 2-3pm, Huxley 140
Abstract: We present a differential system which generalises in degree the Cartan structural equations of Riemannian geometry. We study its main consequences in dimension 3, where the system is given by three 2-forms α0,α1,α2 defined on the contact mani- fold (SM3,θ) which is the 2-sphere tangent bundle of any given oriented Riemannian 3-manifold M 3 . This intrinsic structure of Riemannian geometry relates to Conti- Salamon hypo structures and may be developed as an odd-dimensional version of twistor geometry. We also show the construction, through the same structural equa- tions, of a natural G2 structure on SM4 for any given oriented Riemannian 4-manifold.

### Spring Term 2015

Jonas Hirsch. Boundary regularity of Dirichlet minimizing Q-valued functions Thursday 15th January, 2-3pm, Huxley 642
Abstract: F.J.Almgren introduced a theory of multivalued/ Q-valued functions in his pioneering “big regularity paper”. Q ∈ N, fixed, indicates the number of values the function takes, counting multiplicity. He used the regularity properties of Dirichlet minimizers as an essential tool in his proof of the regularity for mass minimzing intergral currents.
Almgren’s idea motivates the study of the regularity properties of Q-valued functions. I will present an extension of his interior Hölder regularity result up to the boundary.
Firstly I will give a rough explanation of his strategy and a short introduction to the theory of Q-valued functions. Afterwards I will present an overview of the regularity results known so far and some examples, to demonstrate the difficulties one encounters. In the last part I will sketch the proof of the boundary regularity result.

Eleonora Di Nezza. Regularizing properties and uniqueness of the Kaehler-Ricci flow Thursday 22nd January, 2-3pm, Huxley 341
Abstract: Let X be a compact Kaehler manifold. I will show that the Kaehler-Ricci flow, as well as its twisted version, can be run from an arbitrary positive closed current, and that it is immediately smooth in a Zariski open subset of X.
Moreover, if the initial data has positive Lelong number we indeed have propagation of singularities for short time. Finally, if time permits, I will prove a uniqueness result in the case of zero Lelong numbers.
(This is joint work with Chinh Lu)

Nicola Gigli. Nonsmooth differential geometry Thursday 29th January, 2-3pm, Huxley 341
Abstract: I’ll discuss in which sense general metric measure spaces possess a natural first order differential structure and how on spaces with Ricci curvature bounded from below a second order one arises. Objects which will turn out to be well defined in this latter context include: Hessian, covariant/exterior differentiation, connection/Hodge Laplacian and Ricci curvature.

Reto Mueller. The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds Thursday 5th February, 2-3pm, Huxley 341
Abstract: A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy The Nguyen.

Jonathan Luk. Stability and instability of Cauchy horizon in black hole interior Thursday 12th February, 2-3pm, Huxley 341
Abstract: The celebrated strong cosmic censorship conjecture in general relativity in particular predicts that the smooth Cauchy horizons present in the interior of explicit Reissner-Norstr\”om and Kerr black hole solutions are unstable against perturbations. We present recent results regarding both stability (joint with M. Dafermos) and instability (joint with S.-J. Oh) of the smooth Cauchy horizon. The former result shows that for all small perturbations (without symmetry assumptions), the solution possesses a Cauchy horizon such that the metric remains continuous. In particular, if there are singularities in these perturbed spacetimes, they must be so-called weak null singularities.

Anda Degeratu. Analysis on QAC spaces Thursday 19th February, 2-3pm, Huxley 341

Abstract: In this talk I will introduce the class of quasi-asymptotically conical (QAC) manifolds, a less rigid Riemannian formulation of the QALE geometries introduced by Joyce in his study of crepant resolutions of Calabi-Yau orbifolds. Our set-up is in the category of real stratified spaces and Riemannian geometry. Given a QAC manifold, we identify the appropriate weighted Sobolev spaces, for which we prove the finite dimensionality of the null space for generalized Laplacian as well as their Fredholmness.

The methods we use are based on techniques developed in geometric analysis by Grigor’yan and Saloff-Coste, as well as Colding and Minicozzi, and Peter Li. We show that our geometries satisfy the volume doubling property and the Poincar\’e inequality, and we use these properties to analyze the heat kernel behaviour of a generalized Laplacian and to establish Li-Yau type estimates for it. With these estimates we construct a parametrix for our operator and establish our Fredholm results.

This work is joint with Rafe Mazzeo.

Lucas Ambrozio. Static manifolds and Einstein manifolds with edge-cone singularities Thursday 26th February, 2-3pm, Huxley 341
Abstract: A Riemannian manifold is called static when it admits a non-trivial solution to a certain second-order homogeneous overdetermined elliptic equation that naturally appears both in Geometry (e.g, in the problem of prescribing the scalar curvature) and Physics (e.g., in the study of static black-holes).

In this talk, we will discuss some classification results for three-dimensional static manifolds with positive scalar curvature. The main ideia will be to explore the geometry of the (Riemannian) Einstein (n+1)-manifold that can be constructed from every static n-manifold.

Yanir Rubinstein. The degenerate special Lagrangian equation Thursday 5th March, 2-3pm, Huxley 341
Abstract: In joint work with J. Solomon, we introduce the degenerate special Lagrangian equation and develops the basic analytic tools to construct and study its solutions. This equation governs geodesics in the space of positive Lagrangians.

Ben Sharp. Compactness theorems for minimal hypersurfaces with bounded index Thursday 12th March, 2-3pm, Huxley 341
Abstract: I will present a new compactness theorem for minimal hypersurfaces embedded in a closed Riemannian manifold N^{n+1} with n<7. When n=2 and N has positive Ricci curvature, Choi and Schoen proved that a sequence of minimal hypersurfaces with bounded genus converges smoothly and graphically to some minimal limit. A corollary of our main theorem recovers the result of Choi-Schoen and extends this appropriately for all n<7.

Emanuele Spadaro. An improved estimate of the singular set of Dir-minimizing multiple valued functions Thursday 19th March, 2-3pm, Huxley 341
Abstract: In the theory of higher codimension minimal surfaces (in the sense of area minimizing integer rectifiable currents) a prominent role is played by the multiple valued functions minimizing a suitable generalization of the Dirichlet energy, which give the right blowup limits for the analysis of singularities.
In this seminar I will give a brief introduction to this theory and I will present an improved estimate on the Minkowski dimension of the singular set of such functions. This is a joint work with M. Focardi (University of Florence) and A. Marchese (MPI Leipzig).

Hans-Bert Rademacher. Closed geodesics, the free loop space and string topology Thursday 26th March, 2-3pm, Huxley 642
Abstract: Existence results for closed geodesics on compact manifolds can be obtained using equivariant Morse theory on the free loop space. We present an overview on results for Riemannian and Finsler metrics and discuss how string topology can be used to prove resonance statements (joint work with Nancy Hingston).

### Autumn Term 2014

Ezequiel Barbosa. A type of positive mass theorem and applications. Thursday 9th October, 2-3pm, Huxley 341
Abstract: In this talk, we will consider an asymptotically flat manifold (M,g) of dimension n\geq 3 with a noncompact boundary \Sigma, whose geometry at infinity is modeled on a half space R^n_+\subset R^n. To any such manifold (M,g) we associate an asymptotic invariant m_(M,g), which measures the rate of convergence of g toward the flat metric \delta at infinity. This is what we call the mass of (M,g). Under natural positivity conditions on the geometry of (M,g,\Sigma), we prove a positive mass inequality, including a rigidity statement, if either n\leq 7 or n\geq3 and M is spin. We will discuss an application of this result to the global convergence of a certain Yamabe-type flow. Also, we will discuss a relation between this type of positive mass theorem and the classical one.

Lu Wang. A Sharp Lower Bound for the Entropy of Closed Hypersurfaces up to Dimension Six. Thursday 16th October, 2-3pm, Huxley 341
Abstract: Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinkers in R^{n+1}, the entropy is uniquely minimized at the round sphere. They conjectured that, for dimension between 2 and 6, the round sphere minimizes the entropy among all closed smooth hypersurfaces. Joint with Jacob Bernstein, we prove their conjecture via an appropriate weak mean curvature flow. For these dimensions, our approach also gives a new proof of the main result of Colding-Ilmanen-Minicozzi-White and extends its conclusions to compact singular self-shrinkers.

Katrin Leschke. Integrable system methods for minimal surfaces. Thursday 23rd October, 2-3pm, Huxley 341
Abstract: Minimal surfaces, that is surfaces with vanishing mean curvature, are amongst the surface classes best studied and understood. One of the reasons for this is the fact that a minimal surface in 3-space is the real part of a holomorphic function into complex 3-space, and thus the classical notions and facts from Complex Analysis can be used in the study of minimal surfaces. On the other hand, harmonic maps into appropriate spaces give rise to integrable systems. In particular, integrable system methods can be used to investigate surfaces given by a harmonicity condition. For example, constant mean curvature (CMC) surfaces have harmonic Gauss map and the associated family (for non-vanishing mean curvature) has been used to classify all CMC tori as meromorphic functions on an auxiliary Riemann surface given by the associated family, the spectral curve. In this talk, I will explain how some tools from integrable systems, i.e., the associated family and its dressing, give rise to well-known concepts of minimal surfaces. In particular, this indicates that results on minimal surfaces may be special cases of a more general integrable system theory for conformal immersions.

Arick Shao. Unique continuation from infinity for linear waves. Thursday 13th November, 1:45-3pm, Huxley 341
Abstract: We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. These results rely on a new family of geometrically robust Carleman estimates near null cones and on an adaptation of the standard conformal inversion of Minkowski spacetime. Also, the results are nearly optimal in many respects.
This is joint work with Spyros Alexakis and Volker Schlue.

Costante Bellettini. Regularity of semi-calibrated 2-cycles Thursday 20th November, 1:45-3pm, Huxley 341
Abstract: Semi-calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of semi-calibrated currents. We will focus on the case of dimension 2, where it turns out that semi-calibrated cycles are actually pseudo holomorphic. By using an analysis implementation of the algebro-geometric blowing up of a point we study the regularity of semi-calibrated 2-cycles from the point of view of uniqueness of tangent cones and of local smoothness.

Jan Sbierski. Title: A dezornification of the proof of the existence of a maximal Cauchy development for the Einstein equations Thursday 27th November, 1:45-3pm, Huxley 341
Abstract: In 1969, Choquet-Bruhat and Geroch showed that there exists a unique maximal Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In particular, their proof ensures the existence of the maximal development without actually constructing it.
In this talk, we present a proof of the existence of a maximal Cauchy development which avoids the use of Zorn’s lemma and, moreover, provides an explicit construction of the maximal development.

Jose’ Espinar. Title: Escobar type Theorems for fully nonlinear conformal equations on subdomains of the sphere Thursday 4th December, 1:45-3pm, Huxley 341
Abstract: In this talk we will extend Escobar’s classification result [E] to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) and annular domains, both with prescribed constant mean curvature on its boundary. In the latter case, the prescribed constant mean curvature might be different on each boundary component.
[E] J. F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate. Comm. Pure Appl. Math. {\bf 43} (1990), no. 7, 857–883.

Laurent Mazet. Title: Minimal hypersurfaces asymptotic to the Simons cone Thursday 11th December, 1:45-3pm, Huxley 341
Abstract: We prove that, up to similarity, there are at most two miminal hypersurfaces in $\mathbb{R}^{n+2}$ asymptotic to a Simons cone.

There were no seminars on 30th October and 6th November.

### Summer Term 2014

Igor Wigman. Topologies of nodal sets of random band limited functions. Thursday 1st May, 2-3pm, Huxley 140
This work is joint with Peter Sarnak.​
It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

Florent Balacheff. Systolic contact geometry Thursday 8th May, 1-2pm, Huxley 140
Systolic geometry involves a lot of ingredients like algebraic topology, metric geometry or conformal techniques for instance. In this talk, after briefly recall part of this background, we will explain why contact geometry is a natural setting for the study of isosystolic inequalities and the new perspectives it offers. This is joined work with J.C. Alvarez Paiva and K. Tzanev.

Alexander Volkmann. Isoperimetric structure of asymptotically conical manifolds Thursday 15th May, 2-3pm, Huxley 130
We study the isoperimetric structure of Riemannian manifolds that are asymptotic to cones with non-negative Ricci curvature. Specifically, we generalize to this setting the seminal results of G. Huisken and S.-T. Yau on the existence of a canonical foliation by volume preserving stable constant mean curvature surfaces at infinity of asymptotically flat manifolds as well as the results of M. Eichmair and S. Brendle, and M. Eichmair and J. Metzger on the isoperimetric structure of asymptotically flat manifolds. We also make an observation on the isoperimetric cone angle of such manifolds. This result is a natural analogue of the positive mass theorem in this setting.
This is joint work with Otis Chodosh and Michael Eichmair.

Panagiota Daskalopoulos. Ancient Solutions to Geometric Flows Thursday 12th June, 2-3pm, Huxley 130
We will discuss ancient solutions to geometric evolution equations such as the Ricci flow on surfaces, the Mean curvature flow and the Yamabe flow. We will address the classification of such solutions as well as the construction of new examples.

There were no seminars on 22nd, 29th May and 5th, 19th, 26th June.

### Spring Term 2014

Huy The Nguyen. Conformal Immersions of Surfaces into Riemannian Manifolds Thursday 16th January, 3-4pm, Huxley 658

In this talk, we develop the geometry and analysis of conformal immersions into compact/asymptotically flat Riemannian manifolds. We prove an optimal Wente estimate for the Liouville equation and prove a singularity type removal theorem generalising work of Muller-Sverak and Kuwert-Li. We use this to extend a theorem of Brian White to the setting of immersions of surfaces into Riemannian manifolds. Finally, we generalise Helein’s compactness theorem for immersions with bounded total curvature providing optimal constants.

Gustav Holzegel. Linear Stability of the Schwarzschild solution Thursday 23rd January, 3-4pm, Huxley 658
I will talk about recent work joint with M. Dafermos and I. Rodnianski establishing the linear stability of the Schwarzschild solution. Key to the proof is a novel quantity which decouples from the system of gravitational perturbations and satisfies a wave equation, for which decay estimates can be proven. I will also connect the result to the non-linear stability problem.

Panagiotis Gianniotis. Boundary estimates for the Ricci flow Thursday 30th January, 3-4pm, Huxley 658
Shi’s higher order estimates for the curvature and Hamilton’s compactness theorem are essential tools in the study of the singularities of the Ricci flow on complete manifolds. In this talk, I will consider the Ricci flow on manifolds with boundary, with the mean curvature and conformal class of the boundary appropriately controlled. I will present some new higher order estimates valid near the boundary and discuss a compactness result for sequences of Ricci flows.

Eleonora Di Nezza. Monge-Ampère equations on quasi-projective varieties Thursday 6th February, 3-4pm, Huxley 658
We consider X a compact Kaehler manifold and D a divisor in X. We study complex Monge-Ampère equations on the quasi-projective variety X\D, establishing uniform a priori estimates which generalize both Yau’s and Kolodziej’s celebrated estimates. This is a joint work with Hoang-Chinh Lu (CTH, Sweden).

Claude Warnick. Asymptotically anti-de Sitter manifolds Thursday 13th February, 3-4pm, Huxley 658
Asymptotically anti-de Sitter manifolds are the Lorentzian analogue of asymptotically hyperbolic spaces. They arise naturally when considering solutions of Einstein’s equations which have a negative cosmological constant. Much as the hyperbolic disk has a ‘circle at infinity’, these spaces have a cylinder at infinity. I will discuss how the geometric PDE problems on these backgrounds are naturally formulated as initial-boundary value problems with boundary data prescribed on the cylinder at infinity. I will show how the usual L^2 energy estimates can be modified to account of the infinite boundary. If time permits I will present recent work with Holzegel in which we construct dynamical solutions of Einstein’s equations with a negative cosmological constant coupled to a scalar field.

Goncalo Oliveira. Monopoles Thursday 27th February, 3-4pm, Huxley 658
I will explain what 3 dimensional monopoles are and motivate them. Then, for asymptotically conical 3 manifold I will describe how to construct an open set in the moduli space of monopoles.

Tobias Lamm. Global estimates for harmonic maps from surfaces Thursday 6th March, 3-4pm, Huxley 658

Mariel Saez. A Minimax Construction of Embedded Hypersurfaces with Prescribed Constant Mean Curvature in Riemannian Manifolds Thursday 20th March, 3-4pm, Huxley 658
Let M^{n+1} be a closed Riemannian manifold. We prove that there is a constant H_0>0 (depending only on M), such that for every 0<H<H_0 there is an embedded hypersurface \Sigma^n with constant mean curvature H. We also show that \Sigma is smooth except possibly on a set of Hausdorff dimension at most n-7. This is work in progress with Martin Li.

Alix Deruelle. Stability of non compact expanding gradient Ricci solitons Thursday 27th March, 3-4pm, Huxley 658
The Ricci flow, introduced by Hamilton, can be interpreted as a dynamical system on the set of Riemannian metrics of a fixed manifold modulo the action of diffeomorphisms and homotheties. The fixed points of such dynamical system are called Ricci solitons. There are three types according to their lifetime, in particular, expanding Ricci solitons are immortal solutions that can be used to smooth out metric cones. We investigate the stability of such non negatively curved Ricci expanders.

There were no seminars on 20th February and 13th March.

### Autumn Term 2013

Thomas Marquardt. A Neumann free boundary value problem for inverse mean curvature flow Thursday 10th October, 2-3pm, Huxley 341

We consider hypersurfaces with boundary which evolve in the direction of the unit normal with speed equal to the reciprocal of the mean curvature. The boundary condition is of Neumann type, i.e. the evolving hypersurface moves along but stays perpendicular to a fixed supporting hypersurface. In the case where the supporting hypersurface is a convex cone we prove long-time existence for star-shaped initial hypersurfaces of strictly positive mean curvature. In the general case, however, one can not expect the flow to exist for all time. Therefore, we use a level-set approach together with a variational formulation to prove the existence of weak solutions. Furthermore, we indicate the existence of a monotone quantity which is the analog of the Hawking mass for closed hypersurfaces.

Dan Ketover. Degeneration of min-max sequences in 3-manifolds Thursday 10th October, 3-4pm, Huxley 341

The min-max construction of minimal surfaces goes back to Birkhoff who produced closed geodesics on spheres with arbitrary metric. Pitts, using ideas of Almgren proved that the construction gives rise to smooth embedded minimal surfaces in 3-manifolds. Given a Heegaard splitting in a 3-manifold, one has a natural set of sweepouts to consider which generate min-max sequences converging as varifolds to a smooth minimal surface (possibly disconnected, and with multiplicities). Pitts-Rubinstein made a number of conjectures in the 80s about the topology and Morse index of these limiting surfaces. We will give a proof of one of these conjectures about how the sequence may degenerate. The improved convergence we show also gives rise to new genus bounds for min-max limits.

Jeremie Szeftel. The resolution of the bounded L2 curvature conjecture in general relativity Thursday 17th October, 2-3pm, Huxley 341.

In order to locally control a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds on the curvature tensor on a given space-like hypersuface. I will present the proof of this conjecture, which sheds light on the specific nonlinear structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.

Heudson Mirandola. The ADM mass for graphical manifolds Thursday 24th October, 2-3pm, Huxley 341.

By elementary methods we will give explicit formulas for the ADM mass for graphical manifolds of arbitrary dimension and codimension. This will allow us to conclude the positive mass theorem and Penrose inequality for a class of graphical manifolds which includes, for instance, the ones with flat normal bundle. This is a joint work with Feliciano Vitorio.

Alessandro Carlotto. Complete minimal surfaces in asymptotically flat spaces and localized solutions of the Einstein constraint equations Thursday 31st October , 2-3pm, Huxley 341.

To what extent can the global theory of complete minimal surfaces in the Euclidean space can be extended to asymptotically flat 3-manifolds? Starting with the most basic question, we might ask whether complete stable minimal surfaces actually exist in presence of a positive ADM mass, thus considering a generalized Bernstein problem. The answer to this question turns out to be NO if the ambient metric has a nice expansion at infinity (namely if it is asymptotically Schwarschild, in a weak sense), while it is YES in full generality.
The former rigidity result implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. The proof of this theorem is based on a characterization of finite index minimal surfaces, on classical infinitesimal rigidity results by Fischer-Colbrie and Schoen and on the positive mass theorem by Schoen-Yau. More specifically, we also show that in asymptotically flat manifolds a minimal submanifold of codimension one has finite index if and only if it has finitely many ends and each of these is a graph of a function that has a suitable expansion at infinity, in analogy with a classical result by Schoen for Euclidean spaces.
On othe other hand, the latter result (which is joint with Richard Schoen) follows from the construction of a new class of solutions of the Einstein constraint equations that have positive mass but are (exactly) Euclidean on the complement of a cone of arbitrarily small opening angle. This flexibility theorem sharply contrasts various recent scalar curvature rigidity results both in the compact and in the free-boundary setting.
Our main theorems can be extended to analyze marginally outer-trapped surfaces (MOTS) in non time-symmetric initial data sets: we first prove that a non-compact stable MOTS in an initial data set (M,g,k) is conformally diffeomorphic to either the plane C or to the cylinder A and in the latter case infinitesimal rigidity holds. If the data have harmonic asymptotics, the former case is proven to be globally rigid in the sense that the presence of a stable MOTS forces an isometric embedding of (M,g,k) in the Minkowski space-time as a space-like slice.

Mircea Petrache. Singular bundles and regularity of Yang-Mills minimizers in
supercritical dimension
Thursday 7th November, 2-3pm, Huxley 341.

I will present approximation, existence and regularity results for Yang-Mills minimizers in supercritical dimensions, based on a joint project with Tristan Rivière.

The starting point are Uhlenbeck’s results which provided the analytic foundations for Donaldson’s study of Yang-Mills connections on bundles over 4-manifolds. The object of study in that case was the class of Sobolev connections on smooth bundles.

In dimensions 5 and higher the space of Sobolev connections over smooth bundles does not allow to apply the direct methods of the Calculus of Variations to obtain Yang-Mills minimizers. The substitute is a space of weak connections over singular bundles, in which a weak closure result allows constructing Yang-Mills connections by direct minimization. This space is a real measure-theoretic counterpart to singular objects of more algebraic flavour, e. g. coherent reflexive sheaves.

The main tool for the optimal partial regularity result for Yang-Mills minimizers in 5 dimensions is an approximation of weak connections by connections with finitely many topological defects. Such approximation allows to apply a Morrey space analogue of Uhlenbeck’s result, relaxing the approximability hypothesis from previous singularity removal results by Tao-Tian and Meyer-Rivière. We will contrast this new approximation result with approximation results for nonlinear Sobolev maps and for integral currents.

Magdalena Rodríguez. Minimal surfaces in H2xR with finite total curvature and related problems Thursday 14th November, 2-3pm, Huxley 341.

In this talk we will introduce existence and classification results for minimal surfaces in H2xR under several hypothesis. We will also explainhow to construct a counterexample to the Calabi-Yau conjecture for embedded minimal surfaces in H2xR.

Stéphane Sabourau. Sweepouts and volume on Riemannian manifoldsThursday 28th November, 2-3pm, Huxley 341.

The existence of a closed geodesic on every Riemannian two-sphere can be obtained using a minimax principle on the loop space.
This principle extends to the one-cycle space and yields a closed geodesic in this case too.
We will present various curvature-free relationships between the length of this closed geodesic and the area of the sphere, as well as other related results.

Mahir Hadžić . Stability of steady states in the classical Stefan problem. Thursday 5th December , 2-3pm, Huxley 341.

The Stefan problem is a well-known free boundary problem modelling phase transitions. I will explain some recent results on the well-posedness and stability theory in presence and absence of surface tension. I will then show a global stability result in absence of surface tension, thereby explaining a hybrid methodology combining high-order energy methods and quantitative Hopf-type lemmas. This is joint work with Steve Shkoller.

No talks on October 3rd, November 21st, December 12th.

### Summer Term 2013

Hojoo Lee (Korea Institute for Advanced Study). Generalizations of Calabi’s correspondence. Thursday May 9th, 2-3pm, Huxley 140.
We generalize Calabi’s correspondence between minimal surfaces in Euclidean space and maximal surfaces in Lorentz-Minkowski space-time.

> May 10: Geometry Day IV at King’s College London.

Special lecture series by Simon Donaldson and Song Sun: Kähler-Einstein metrics on Fano manifolds. Huxley 140.

• 16th May, 2-4pm, Part I:
Lecture A. General background in Kahler geometry. The Futaki invariant and the definition of K-stability. Strategy of proof of the main result. (Arxiv:1210.7494)
Lecture B. Review of Gromov-Hausdorff convergence and Cheeger-Colding theory. The Hörmander technique in complex analysis. Complex algebraic structures on GH limits. (Arxiv:1206.2609)
• 23rd May, 2-4pm, Part II:
Lecture A. Foundations of theory of Kahler metrics with cone singularities along a divisor. Schauder estimates. Approximation by smooth metrics with positive Ricci curvature. (Arxiv:1102.1196, Arxiv:1211.4566)
Lecture B. Extension of results in Lecture 1B to the case of cone singularities. (Arxiv:1212.4714)
• 6th June, 2-4pm Part III:
Lecture A. Continuation of 2B. Regularity theory for limits. (Arxiv:1212.4714, Arxiv:1302.0282)
Lecture B. Pluripotential theory and an extension of Matsushima’s Theorem. Completion of proof. (Arxiv:1302.0282)

Dan Knopf (University of Texas at Austin). Type-II singularities of Ricci flow. Wednesday May 29th, 2-3pm, Huxley 140.
We will discuss recent progress (by the speaker and collaborators) in determining what rates of finite-time singularity formation are possible for either compact or noncompact solutions of Ricci flow, and in constructing degenerate neckpinch solutions with prescribed asymptotic behaviors.

Special event: Warwick-Imperial-Cambridge Geometric Analysis Seminar. Thursday May 30th, Huxley 139.
Schedule:

• 11:00-11:30 Reception (Maths Common Room, Huxley 549)
• 11:30-12:20 Frank Pacard (Ecole Polytechnique)
• 12:20-15:00 Lunch
• 16:00-16:30 Coffee break (Maths Common Room, Huxley 549)
• 16:30-17:30 Tom Ilmanen (ETH Zürich)

There will also be a dinner at 20:00 – venue to be announced.

> May 31: Geometry and Topology Days (Event 3: Geometric Analysis) at University College London.

No talks on June 13th, 20th and 27th.

### Spring Term 2013

Willie Wong (EPFL Lausanne). A measure of closeness-to-Kerr-Newman for stationary electro-vacuum space-times and applications. Thursday March 21th, Huxley 130, 2.00-3.00 pm.
We describe a local covariant characterisation of the Kerr-Newman solutions in general relativity (among stationary solutions to Einstein’s equation coupled to an electromagnetic field) and write down some geometric identities involving this quantity. Then we sketch how these identities allow us to generalise and improve the conditional black hole uniqueness results in the smooth category originally due to Ionescu-Klainerman and Alexakis-Ionescu-Klainerman. This is joint work with Pin Yu.

Ari Laptev (Imperial College and Institut Mittag-Leffler). Spectral properties of Schrödinger operators on compact manifolds: rigidity, flows, interpolation and spectral estimates. Thursday March 14th, Huxley 130, 2.00-3.00 pm.
This talk is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.

Melanie Rupflin (Max Planck Potsdam). Teichmueller harmonic map flow. Thursday March 7th, Huxley 130, 2.00-3.00 pm.
We introduce and discuss a new geometric flow, the Teichmueller harmonic map flow, which is designed to evolve closed surfaces in general target manifolds towards (possibly branched) minimal surfaces.We discuss the questions of existence and regularity of solutions of this flow and explain why global solutions of Teichmueller harmonic map flow converge to critical points of the area functional with similar topological properties as the initial data. This is joint work with Peter Topping.

Anda Degeratu (University of Freiburg). Witten spinors on nonspin manifolds. Thursday February 28th, Huxley 130, 2.00-3.00 pm.
Unlike a 3-dimensional manifold, a higher dimensional manifold need not be spin. On an oriented Riemannian manifold the obstruction to having a spin structure is given by the second Stiefel-Whitney class. I will show that even when this obstruction does not vanish, it is still possible to define a notion of singular spin structure and associated singular Dirac operator. Then, modeling on Witten’s proof of the Positive Mass Theorem, I will define the notion of Witten spinor on an asymptotically flat nonspin manifold, show their existence and describe their properties.

Esther Cabezas-Rivas (University of Münster). A generalization of Gromov’s almost flat manifold theorem. Thursday February 14th, Huxley 130, 2.00-3.00 pm.
Taking as a starting point a question by John Lott about the vanishing of the $\hat A$-genus for spin almost-non-negatively curved manifolds, we conjecture that an almost-non-negatively curved manifold is either conformally equivalent to a manifold with positive scalar curvature or it is finitely covered by a Nilmanifold. In the way to prove such a claim, we found a generalization of Gromov’s almost flat manifold theorem where $L^\infty$ bounds for the curvature are relaxed to mixed curvature bounds. During the talk, we will give the precise statement of our theorem and a detailed sketch of the proof. This is a joint work with Burkhard Wilking.

Jake Solomon (Hebrew University of Jerusalem). The space of positive Lagrangian submanifolds. Thursday February 7th, Huxley 130, 2.00-3.00 pm.
A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I’ll explain how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in the context of the Kobayashi-Hitchin correspondence.

Qian Wang (Oxford University). Rough solutions of Einstein vacuum equations in CMCSH gauge. Thursday January 24th, Huxley 130, 2.00-3.00 pm.
We consider very rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local well-posedness result in $H^s, s>2$. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric $\textbf{g}$, we manage to implement the commuting vector field approach to prove Strichartz estimate for the geometric wave equation $\Box_\textbf{g} \phi=0$ directly.

Lorenzo Foscolo (Imperial College London). A gluing construction for periodic monopoles. Thursday January 10th, Huxley 130, 2.00-3.00 pm.
Periodic monopoles are solutions to the Bogomolny equation (the dimensional reduction of the anti-self-duality equation to 3 dimensions) on $\mathbb{R}^2 \times \mathbb{S}^1$. Using arguments from physics, Cherkis and Kapustin provided strong evidence that 4-dimensional moduli spaces of periodic monopoles with singularities yield examples of gravitational instantons (i.e. complete hyperkähler 4-manifolds with decaying curvature at infinity) of type ALG. We will present a gluing construction aimed to
(i) prove that periodic monopoles (with singularities) exist;
(ii) provide a description of a neighbourhood of the end of the moduli space which allows to calculate the asymptotics of the metric and verify Cherkis and Kapustin’s predictions.

### Autumn Term 2012

Reto Müller (Imperial College London). Łojasievicz-Simon inequalities for the Ricci flow and applications. Thursday December 13th, Huxley 130, 2.00-3.00 pm.
In this talk, we discuss different stability, instability, and uniqueness questions for the Ricci flow that can be attacked with the help of Łojasievicz-Simon inequalities for Perelman’s monotone functionals $\mathcal{F}$ and $\mathcal{W}$.

Jason Cantarella (University of Georgia). The geometry of random (closed) space polygons. Thursday November 29th, Huxley 130, 2.00-3.00 pm.
Here are some natural questions from statistical physics: What is the expected shape of a ring polymer with $n$ monomers in solution? What is the expected knot type of the polymer? The radius of gyration? Numerical simulations are absolutely required to make progress on these questions, but pose some interesting challenges for geometers.
We wish to sample the space of closed $n$-gons in 3-space, which is a nonlinear submanifold of the larger space of open $n$-gons. To sample equilateral polygon space, current algorithms use a Markov process which randomly “folds” polygons while preserving closure and edgelengths. Such algorithms are expected to converge in $O(n^3)$ time.
The main point of this talk is that a much better sampling algorithm is available if we widen our view to the space of $n$-gons in three dimensional space of fixed total length (rather than fixed edgelengths). We describe a natural probability measure on length 2 $n$-gon space pushed forward from the standard measure on the Stiefel manifold of 2-frames in complex $n$-space using methods from algebraic geometry. We can directly sample the Stiefel manifold in $O(n)$ time, allowing us to generate closed polygons using the pushforward map.
We will give some explicit computations of expected values for geometric properties for such random polygons, discuss their topology, and compare our theorems to numerical experiments using our sampling algorithm. The talk describes joint work with Malcolm Adams (University of Georgia, USA), Tetsuo Deguchi (Ochanomizu University, Japan), and Clay Shonkwiler (University of Georgia, USA).

Jacob Bernstein (University of Cambridge). A curious variational property of classical minimal surfaces. Thursday November 22nd, Huxley 130, 2.00-3.00 pm.
Let $\Sigma$ be a nowhere umbilic classical minimal surface in $\mathbb{R}^3$. We observe that the induced metric, $g$, on $\Sigma$ may be conformally deformed – in an explicit manner – to a smooth metric $\hat{g}$ which is a critical point of a natural geometric functional $\mathcal{E}$. The diffeomorphism invariance of $\mathcal{E}$ gives rise
to a conservation law $T$. We characterize several important model surfaces in terms of $T$. Time permitting, the KdV equation will make an unexpected guest appearance. This is joint work with T. Mettler.

Giuseppe Tinaglia (King’s College London). The geometry of constant mean curvature surfaces embedded in $\mathbb{R}^3$. Thursday November 15th, Huxley 130, 2.00-3.00 pm.
In this talk I will discuss recent results on the geometry of constant mean curvature ($H\neq 0$) surfaces embedded in $\mathbb{R}^3$. Among other things I will prove radius and curvature estimates for simply connected surfaces embedded in $\mathbb{R}^3$ with constant mean curvature. It follows from the radius estimate that the only complete, simply connected surface embedded in $\mathbb{R}^3$ with constant mean curvature is the round sphere. This is joint work with Bill Meeks.

Mike Munn (University of Missouri and University of Warwick). Metric perspectives of the Ricci flow. Thursday November 8th, Huxley 130, 2.00-3.00 pm.
Motivated by a characterization of the super Ricci flow given by McCann-Topping, we introduce the notion of a super Ricci flow for a family of distance metrics defined on the disjoint union of two smooth Riemannian manifolds, $M_1$ and $M_2$, evolving by the Ricci flow. In particular, we show that this super Ricci flow property holds when the distance between points in $M_1$ and $M_2$ evolves by the heat equation. We also discuss possible applications and examples. This is joint work with Sajjad Lakzian.

Felix Schulze (Free University Berlin). The half-space property and entire positive minimal graphs in $M \times \mathbb{R}$. Thursday November 1st, Huxley 130, 2.00-3.00 pm.
We show that a properly immersed minimal hypersurface in $M \times \mathbb{R}^{+}$ equals some $M \times \{c\}$ when $M$ is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. If on the other hand, $M$ has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over $M$. This is joint work with H. Rosenberg and J. Spruck.

Robert Kusner (University of Massachusetts). The space of CMC surfaces. Thursday October 25th, Huxley 130, 2.00-3.00 pm.
Understanding the space of all “soap bubbles” – that is, complete embedded constant mean curvatures (CMC) surfaces in $\mathbb{R}^3$ – is a central problem in geometric analysis. These CMC surfaces are highly transcendental objects; the topology and smooth structure of their moduli spaces are understood only in some special cases. In this talk we’ll describe the formal “Lagrangian embedding” of CMC moduli space into the “space of asymptotes” and discuss where this is smooth, namely, at a surface with no nontrivial square-integrable Jacobi fields. This nondegeneracy condition has now been established for all coplanar CMC surfaces of genus 0. These coplanar CMC surfaces have a surprising connection with complex projective structures and holomorphic quadratic differentials on $\mathbb{C}$ obtained by taking the Schwarzian of the developing map for the projective structure. This lets us explicitly work out the smooth topology of their moduli spaces.

Sergio Almaraz (Universidade Federal Fluminense & Imperial College London). Convergence of scalar-flat metrics on manifolds with boundary under the Yamabe flow. Thursday October 18th, Huxley 130, 2.00-3.00 pm.
In this talk I will discuss a convergence theorem for a Yamabe-type flow on manifolds with boundary. This is a flow that evolves conformal scalar-flat metrics according to equations envolving the boundary mean curvature. Convergence to a scalar-flat metric with constant boundary mean curvature is established assuming a positive mass theorem or a genetric condition.

Jan Metzger (University of Potsdam). Minimizers of the Willmore functional with prescribed area. Thursday October 11th, Huxley 130, 2.00-3.00 pm.
In this talk I will discuss joint work with Tobias Lamm on the Willmore functional of immersed surfaces in Riemannian manifolds subject to a small area constraint. We show that for small enough area there are smooth minimizers that concentrate near points of maximal scalar curvature.

Gustav Holzegel (Imperial College London). The black hole stability problem. Thursday October 4th, Huxley 130, 2.00-3.00 pm.
After an introduction to the Black Hole Stability Problem, I will discuss two recent results: For the first, we introduce a class of dynamical spacetimes, so-called “ultimately Schwarzschildean spacetimes”, which converge (at a certain rate in time) to a fixed Schwarzschild spacetime. The main result will establish improved estimates for the Ricci-coefficients and curvature-components in this class of spacetimes. The relevance of these estimates for the black hole stability problem will also be discussed.
The second result constructs explicitly a large class of dynamical vacuum spacetimes which converge to a fixed Schwarzschild-, or more generally, Kerr-spacetime. Here the solutions are obtained in the context of a scattering problem with data prescribed on the event horizon and null-infinity. This is joint work with M. Dafermos and I. Rodnianski.

### Summer Term 2012

Jingyi Chen (UBC Vancouver). Longtime existence of Lagrangian mean curvature flow for entire Lipschitz graphs. Friday June 22nd, Huxley 140, 3.00-4.00 pm.
We prove longtime existence and estimates for solutions to fully nonlinear parabolic equation
with $C^{1,1}$ initial data either whose Hessian has eigenvalues between -1 and 1 or it is convex. This is joint work with Albert Chau and Yu Yuan.

André Neves (Imperial College). Min-max theory, Willmore conjecture, and the energy of links. Thursday June 21st, Huxley 213 (Clore LT), 1.00-3.15 pm.
In 1965, Willmore conjectured that the integral square of the mean curvature is never smaller than $2\pi^2$. I will explain how to prove this conjecture using the min-max theory for minimal surfaces. This is joint work with Fernando Marques (IMPA).
In 1994, Freedman-He-Wang conjectured that the Möbius energy of non-trivial links is never smaller than $2\pi^2$. I will explain how to prove this conjecture using the min-max theory for minimal surfaces. This is joint work with Fernando Marques (IMPA) and Ian Agol (UC Berkely).

Cristiano Spotti (Imperial College). Deformations of singular KE Del Pezzo surfaces and applications to moduli spaces. Thursday June 14th, Huxley 213 (Clore LT), 1.00-2.00 pm.
Given a Kähler-Einstein (KE) Del Pezzo orbifold, it is interesting to study how the problem of the existence of KE metrics behaves under (partial)-smoothings of the singularities. In this talk we will show what happens in the case of nodal Del Pezzo surfaces with discrete automorphism group and we will describe the expected general picture. Finally we will discuss how this deformation theory can be useful to study compactified moduli spaces of KE Del Pezzo surfaces.

Hans-Joachim Hein (Imperial College). A local regularity theorem for the Ricci flow. Thursday June 7th, Huxley 140, 1.00-2.00 pm.
I will explain the main result of a recent joint work with Aaron Naber: If a point in a given Ricci flow spacetime has small entropy, then a definite parabolic neighborhood of this point enjoys definite sectional curvature bounds. Unlike the well-known cases of the Einstein, minimal surface, or mean curvature flow equations, the main difficulty here is to prove that small entropy at a single point implies small entropy in some definite neighborhood. To overcome this problem, we introduce new logarithmic Sobolev inequalities for the adjoint heat kernel measure along the Ricci flow.

Jason Lotay (University College London). Deforming $G_2$ conifolds. Thursday May 24th, Huxley 140, 1.00-2.00 pm.
Two natural classes of $G_2$ manifolds are those which either have non-compact ends asymptotic to cones or have isolated conical singularities. Examples of the former are given by the ﬁrst complete examples of $G_2$ manifolds due to Bryant and Salamon, and the latter play an important role in M-Theory. By the fundamental work of Joyce, a compact $G_2$ manifold $M$ has a smooth moduli space of deformations of dimension $b^3(M)$. I will describe a natural extension of this result to the two aforementioned types of $G_2$ conifolds. In particular, I will show the stark contrast between the deformation theories in each case and give some applications. This is joint work with S. Karigiannis.

Diarmuid Crowley (Max Planck Institute, Bonn). On classifying simply connected 7-manifolds. Thursday May 17th, Huxley 140, 1.00-2.00 pm.
Dimension 7 has a rich history and an active present in both differential topology and geometry. In this talk I give an over view of smooth classification results for simply connected 7-manifolds illustrated by examples of geometric interest. A persistent challenge for 7-manifold topology is to find intrinsic calculations of invariants without using a co-bounding 8-manifolds and global analysis often meets this challenge. For example, with Goette we showed recently how to use the $\eta$-invariants of the Dirac operator twisted by quaternionic line bundles to define interesting new intrinsic invariants for spin 7-manifolds.

Special event: Warwick-Imperial-Cambridge Geometric Analysis Seminar. Thursday May 10th, in Warwick.

• Fernando Codá Marques (IMPA)
• Francesco Maggi (Florence/UT-Austin)
• Sergiu Klainerman (Princeton)

Details can be found here: http://go.warwick.ac.uk/mathsevents.

Brian Krummel (University of Cambridge). The regularity and existence of branched minimal submanifolds. Thursday May 3rd, Huxley 658, 1.30-2.30 pm.
Multivalued functions arise naturally in the study of the branch set of branched minimal immersions. Simon and Wickramasekera (2007) showed how to construct a large class of multivalued solutions in $C^{1,\mu}$ to the Dirichlet problem for the minimal surface equation provided the boundary data satisfied a k-fold symmetry condition. I will extend their existence result, which was specific to the minimal surface equation, to show that there exists multivalued solutions in $C^{1,\mu}$ to other elliptic equations and to elliptic systems that preserve the $k$-fold symmetry condition. I also will show that the branch sets of the minimal hypersurfaces they constructed are real analytic submanifolds, which involves proving a general regularity result for multivalued solutions to elliptic equations.

### Spring Term 2012

Ben Sharp (Warwick University). Compensation phenomena arising in Geometric PDE. Thursday March 22nd, Huxley 658, 1.30-2.30 pm.
We will consider systems of critical PDE which exhibit compensation properties allowing for a regularity theory, first introduced by Rivière ’06 and Rivière-Struwe ’09. These PDE describe, in particular, harmonic maps between Riemannian manifolds and also conformal immersions of surfaces with bounded Willmore energy.
We shall see that these systems can be seen as divergence type problems allowing one to change frame and get around their critical nature, and we will describe some recent improvements in the regularity of such solutions. Some of this work is joint with Peter Topping.

Olivier Biquard (École Normale Paris). Desingularization of Einstein orbifolds. Thursday March 15th, Huxley 658, 1.30-2.30 pm.
I study the problem of desingularization of real Einstein 4-orbifolds with Kleinian singularities, in the simplest case of A1 singularities. An obstruction that is not present in the Kähler case appear. I prove a desingularization theorem in some cases when the obstruction vanishes. Finally, I give some consequences on the problem of finding asymptotically hyperbolic Einstein metrics with given boundary at infinity.

Yohsuke Imagi (Kyoto University). A uniqueness theorem for gluing special Lagrangian submanifolds. Thursday March 8th, Huxley 658, 1.30-2.30 pm.
Special Lagrangian submanifolds are area minimizing Lagrangian submanifolds discovered by Harvey and Lawson. There is no obstruction to deforming compact special Lagrangian submanifolds by a theorem of Mclean. It is however difficult to understand singularities of special Lagrangian submanifolds (varifolds). Joyce has studied isolated singularities with multiplicity one smooth tangent cones. Suppose that there exists a compact special Lagrangian submanifold M of dimension three with one point singularity modelled on the Clliford torus cone. We may apply the gluing technique to $M$ by a theorem of Joyce. We obtain then a compact non-singular special Lagrangian submanifold sufficiently close to $M$ as varifolds in Geometric Measure Theory. The main result of this talk is as follows: all special Lagrangian varifolds sufficiently close to $M$ are obtained by the gluing technique.
The proof is similar to that of a theorem of Donaldson in the Yang-Mills theory. One first proves an analogue of Uhlenbeck’s removable singularities theorem in the Yang-Mills theory. One uses here an idea of a theorem of Simon, who proved the uniqueness of multiplicity one tangent cones of minimal surfaces. One proves next the uniqueness of local models for desingularizing $M$ (see above) using symmetry of the Clifford torus cone. These are the main parts of the proof.

Paul Laurain (Université Paris Diderot). Surfaces with constant mean curvature in a Riemannian manifold of dimension 3. Thursday March 1st, Huxley 658, 1.30-2.30 pm.
The surfaces with constant mean curvature (cmc) in a spacelike hypersurface are geometrically and physically very interesting, as shown in [Huisken-Yau-96] or in the beautiful thesis of H.L. Bray. However, the purpose of this talk is not to develop the physical properties of cmc but to see on an example what are the analytical difficulties encountered when studying these surfaces.
In fact, we will show how to study the cmc in terms of partial differential equations in order to derive geometric properties. We emphasize in particular the key difficulties generated by the conformal invariance of the problem as the phenomena of concentration and we will show how the structure of the equation helps us to overcome them.

Vlad Moraru (Warwick University). Splitting of 3-manifolds and rigidity of area-minimising surfaces. Thursday February 23rd, Huxley 658, 1.30-2.30 pm.
I will prove an area-comparison theorem for certain totally geodesic surfaces in 3-manifolds with lower bound on the scalar curvature. I will use this comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bound on the scalar curvature. This is joint work with Mario Micallef.

Peter Hornung (Max Planck Leipzig). Analysis of intrinsically flat surfaces and the Willmore functional. Thursday February 16th, Huxley 658, 1.30-2.30 pm.
We present some recent results about intrinsically flat surfaces with finite Willmore energy. We also study the restriction of the Willmore functional to this class of surfaces, and we present a regularity result for minimizers of this constrained functional.

Simon Blatt (Warwick University). Analysis of O’Hara’s knot energies. Thursday February 9th, Huxley 658, 1.30-2.30 pm.
All of us know how hard it can be to decide whether the cable spaghetti lying in front of us is really knotted or whether the knot vanishes into thin air after pushing and pulling at the right strings. In this talk we approach this problem using gradient flows of a family of energies introduced by O’Hara in 1991-1994. We will see that this allows us to transform any closed curve into a special set of representatives – the stationary points of these energies – without changing the type of knot. We prove longtime existence and smooth convergence to stationary points for these evolution equations.

Robert Haslhofer (ETH Zürich). Singularities in 4d Ricci flow. Thursday February 2nd, Huxley 658, 1.30-2.30 pm.
In this talk, we discuss the formation of singularities in higher-dimensional Ricci flow without pointwise curvature assumptions. We show that the space of singularity models with bounded entropy and locally bounded energy is orbifold-compact in arbitrary dimensions. In dimension four, a delicate localized Gauss-Bonnet estimate even allows us to drop the assumption on energy in favor of (essentially) an upper bound for the Euler characteristic. We will also see how these results are part of a larger project exploring high curvature regions in 4d Ricci flow. This is all joint work with Reto Müller.

Andrea Mondino (Scuola Normale Pisa). The Willmore and other $L^2$ curvature functionals in Riemannian manifolds. Thursday January 26th, Huxley 658, 1.30-2.30 pm.
Given an immersion $f$ of a surface $\Sigma$ into $\mathbb{R}^3$, called $H$ the mean curvature of the immersion, the Willmore functional is defined as the $L^2$ norm of the mean curvature up to a factor: $W(f):=\frac{1}{4}\int H^2$. If we consider immersions in a Riemannian manifold $(M,g)$ there are many possible generalizations of the Willmore functional $W$; in the seminar we will speak about these generalizations and study the existence of minimizers and critical points of the corresponding functionals under curvature conditions on the ambient manifold $(M,g)$. The topic has links with general relativity, string theory, biology, nonlinear elasticity theory etc.

Thomas Walpuski (Imperial London). $G_2$-instantons on generalised Kummer constructions. Thursday January 19th, Huxley 658, 1.30-2.30 pm.
I will discuss a method to construct $G_2$-instantons on $G_2$-manifolds arising from Joyce’s generalised Kummer construction. After that I will explain when (and why) this construction is expected to give a complete description of the moduli space of $G_2$-instantons. Time permitting I will talk about potential applications to computing a conjectural $G_2$ Casson invariant.

Simon Donaldson (Imperial London). Normalised energy and singular limits in Riemannian geometry. Thursday January 12th, Huxley 130, 2.00-3.00 pm.
The talk will describe some joint work with X-X Chen. An important result of Cheeger, Colding and Tian asserts, roughly speaking, that the formation of a codimension-k singularity in a limit of Riemannian manifolds with a bound on the Ricci tensor requires that the curvature be not too small in $L^{k/2}$. We give a new proof of this which also extends the scope of the result. A central idea in the proof is the use of scaling and the normalised “energy” of balls. The arguments are of a relatively elementary nature and I will attempt to make the talk accessible to a general mathematical audience.