Search

In this talk, we shall describe some topological rigidity and its relationship with positive scalar curvature. Precisely, we will present a proof that a complete contractible 3-manifold with positive scalar curvature is homeomorphic to the Euclidean 3-space. We will furthermore explain the interplay between, minimal surfaces, scalar curvature and the topology at infinity.

We introduce various notions of convergence of Riemannian manifolds and metric spaces. We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We close the course by presenting methods and theorems that may be applied to prove these open questions including older techniques developed with Lakzian, with Huang and Lee, and with Portegies. I will also present key new results of Allen and Perales. Students and postdocs interested in working on these problems will be formed into teams. For a complete list of papers about intrinsic flat convergence see: https://sites.google.com/site/intrinsicflatconvergence/

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

The goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the celebrated Lott-Sturm-Villani theory of CD(K,N) metric measure spaces. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics (with respect to a suitable Lorentzian Wasserstein distance) of probability measures. The smooth Lorentzian setting was previously investigated by McCann and Mondino-Suhr.After recalling the general setting of Lorentzian synthetic spaces (including remarkable examples fitting the framework), I will discuss some basics of optimal transport theory thereof in order to define "timelike Ricci curvature bounded below and dimension bounded above'' for a (possibly non-smooth) Lorentzian space. The notion of "timelike Ricci curvature bounded below and dimension bounded above'' for a (possibly non-smooth) Lorentzian space is stable under a suitable weak convergence of Lorentzian synthetic spaces, giving a glimpse on the strength of the proposed approach.As an application of the optimal transport approach to timelike Ricci curvature lower bounds, I will discuss an extension of the Hawking's Singularity Theorem (in sharp form) to the synthetic setting.

We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the Bochner formula and basics of Ricci flow.

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how to establish existence via elliptic regularization. We will furthermore discuss tangent flows and regularity, and the interaction of Brakke flows with the level set flow. Time permitting, we will give an outlook on recent developments, including the proof of the mean convex neighborhood conjecture by Choi-Haslhofer-Hershkovits/Choi-Haslhofer-Hershkovits-White as well as progress towards establishing the existence of generic mean curvature flow.

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how to establish existence via elliptic regularization. We will furthermore discuss tangent flows and regularity, and the interaction of Brakke flows with the level set flow. Time permitting, we will give an outlook on recent developments, including the proof of the mean convex neighborhood conjecture by Choi-Haslhofer-Hershkovits/Choi-Haslhofer-Hershkovits-White as well as progress towards establishing the existence of generic mean curvature flow.

I will present a joint work with G. Carron and I. Mondello where we study Kato limit spaces. These are metric measure spaces obtained as Gromov-Hausdorff limits of smooth n-dimensional Riemannian manifolds with Ricci curvature satisfying a uniform Kato-type condition. In this context, strictly wider than the ones of Ricci limit spaces (where the Ricci curvature satisfies a uniform lower bound) and Lp-Ricci limit spaces (where the Ricci curvature is uniformly bounded in Lp for some p>n/2), we extend classical results of Cheeger, Colding and Naber, like the fact that under a non-collapsing assumption, every tangent cone is a metric measure cone. I will present these results and explain how we rely upon a new heat-kernel based almost monotone quantity to derive them.

The study of positive scalar curvature on noncompact manifolds has seen significant progress in the last few years. A major role has been played by Gromov's results and conjectures, and in particular the idea to use surfaces of prescribed mean curvature (as opposed to minimal surfaces). Having the classic positive mass theorem of Schoen-Yau in mind, we describe a new positive mass theorem for manifolds that allows for possibly non asymptotically flat ends, points of incompleteness, and regions negative scalar curvature. The proof is based on surfaces with prescribed mean curvature, and gives an alternative proof of the Liouville theorem conjectured by Schoen-Yau, which was recently proved by Chodosh-Li. This is joint with R.Unger and S-T. Yau.

We study the noncollapsed singularity formation of Einstein 4-manifolds. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result of a gluing-perturbation procedure that we develop. This sheds light on the structure of the moduli space of Einstein 4-manifolds near its boundary and lets us show that spherical and hyperbolic orbifolds (which are synthetic Einstein spaces) cannot be GH-approximated by smooth Einstein metrics. New obstructions specific to the compact situation moreover raise the question of whether or not a sequence of Einstein 4-manifolds degenerating while bubbling out gravitational instantons has to be Kähler-Einstein.

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how to establish existence via elliptic regularization. We will furthermore discuss tangent flows and regularity, and the interaction of Brakke flows with the level set flow. Time permitting, we will give an outlook on recent developments, including the proof of the mean convex neighborhood conjecture by Choi-Haslhofer-Hershkovits/Choi-Haslhofer-Hershkovits-White as well as progress towards establishing the existence of generic mean curvature flow.

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

I will describe work in progress, parts of which are joint with Marcelo Alves. Let L be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space which have ideal boundary equal to L, and in this way obtain a knot invariant. In other words the number of minimal surfaces doesn’t depend on the isotopy class of the link. These counts of minimal surfaces can be organised into a two-variable polynomial which is perhaps a known polynomial invariant of the link, such as HOMFLYPT.“Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how this “minimal surface polynomial" can be seen as a Gromov-Witten invariant for the twistor space of hyperbolic 4-space. This leads naturally to a new class of infinite-volume 6-dimensional symplectic manifolds with well behaved counts of J-holomorphic curves. This gives more potential knot invariants, for knots in 3-manifolds other than the 3-sphere. It also enables the counting of minimal surfaces in more general Riemannian 4-manifolds, besides hyperbolic space.

Over the last fifty years, the problem of finding sharp upper bounds for area-normalized Laplacian eigenvalues on closed surfaces has attracted the attention of many geometers, due in part to connections to the study of sphere-valued harmonic maps and minimal immersions. In this talk, I'll describe a series of results which shed new light on this problem by relating it to the variational theory of the Dirichlet energy on sphere-valued maps. Recent applications include new (H^{-1}-)stability results for the maximization of the first and second Laplacian eigenvalues, and a proof that metrics maximizing the first Steklov eigenvalue on a surface of genus g and k boundary components limit to the \lambda_1-maximizing metric on the closed surface of genus g as k becomes large (in particular, the associated free boundary minimal surfaces in B^{N+1} converge as varifolds to the associated closed minimal surface in S^N). Based on joint works with Mikhail Karpukhin, Mickael Nahon and Iosif Polterovich.

Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed, similar to Ricci Flow in the intrinsic setting, to have the potential to serve as a tool to approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states that a generic mean curvature flow should have only spherical and cylindrical singularities. As a first step in this direction Colding-Minicozzi have shown in fundamental work that spheres and cylinders are the only linearly stable singularity models. As a second step toward Huisken's conjecture we show that mean curvature flow of generic initial closed surfaces in R^3 avoids asymptotically conical and non-spherical compact singularities. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact self-similarly shrinking solutions. This is joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.

We introduce various notions of convergence of Riemannian manifolds and metric spaces. We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We close the course by presenting methods and theorems that may be applied to prove these open questions including older techniques developed with Lakzian, with Huang and Lee, and with Portegies. I will also present key new results of Allen and Perales. Students and postdocs interested in working on these problems will be formed into teams. For a complete list of papers about intrinsic flat convergence see: https://sites.google.com/site/intrinsicflatconvergence/

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how to establish existence via elliptic regularization. We will furthermore discuss tangent flows and regularity, and the interaction of Brakke flows with the level set flow. Time permitting, we will give an outlook on recent developments, including the proof of the mean convex neighborhood conjecture by Choi-Haslhofer-Hershkovits/Choi-Haslhofer-Hershkovits-White as well as progress towards establishing the existence of generic mean curvature flow.

We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the Bochner formula and basics of Ricci flow.

Summer school 2021. The theory of non smooth spaces with lower Ricci Curvature bounds has undergone huge developments in the last thirty years. On the one hand the impetus came from Gromov’s precompactness theorem and the Cheeger-Colding theory of Ricci limit spaces. On the other hand “synthetic” theories of lower Ricci bounds have been developed, based on semigroup tools (the Bakry-Émery theory) and on Optimal Transport (the Lott-Sturm-Villani theory).The Cheeger-Colding theory did not consider manifolds with boundary, while in the synthetic framework even understanding what is a good definition of boundary is a challenge.The aim of this talk is to present some recent results obtained in collaboration with E. Bruè (IAS, Princeton) and A. Naber (Northwestern University) about regularity and stability for boundaries of spaces with lower Ricci Curvature bounds.

We introduce various notions of convergence of Riemannian manifolds and metric spaces. We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature. We close the course by presenting methods and theorems that may be applied to prove these open questions including older techniques developed with Lakzian, with Huang and Lee, and with Portegies. I will also present key new results of Allen and Perales. Students and postdocs interested in working on these problems will be formed into teams. For a complete list of papers about intrinsic flat convergence see: https://sites.google.com/site/intrinsicflatconvergence/

We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the Bochner formula and basics of Ricci flow.