Talks (titles and abstracts)
Tobias H. Colding: tba
Guido De Philippis: tba
Giada Franz: Construction and properties of free boundary minimal surfaces via min-max
A free boundary minimal surface (FBMS) in a three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold. It is natural to ask about the existence of FBMS (in a given ambient manifold) and their properties (topology, area, Morse index, etc.).
In this talk, we will analyze these questions through the lens of Simon-Smith variant of Almgren-Pitts min-max theory. More precisely, we will see how this method allows the construction of FBMS with prescribed properties (symmetry, topology, Morse index, etc.), by presenting new developments and discussing the limits and perspectives of this approach.
Ailana Fraser: tba
Lan-Hsuan Huang: Structure theory of Einstein manifolds with boundary
We discuss results on the structure of compact Einstein manifolds in terms of the conformal boundary metric and boundary mean curvature. In three dimensions, we confirm M. Anderson's conjecture, showing that the map from the space of Einstein metrics to such boundary data is generically a local diffeomorphism. Additionally, we discuss analogous results in dimensions greater than three for Ricci flat manifolds or Einstein manifolds with a negative constant, assuming the non-degenerate boundary condition. This talk is based on joint work with Zhongshan An (University of Michigan).
Paul Laurain: tba
Chao Li: tba
Andrea Malchiodi: tba
Christos Mantoulidis: Improved generic regularity for minimizing hypersurfaces
Lorenzo Mazzieri: tba
Marco Méndez Guaraco: tba
William Minicozzi: tba
Alessandro Pigati: tba
Antonio Ros: tba
Thomas Schick: Rigidity of positive scalar curvature under low regularity
In this talk we discuss extremality and rigidity properties of positive scalar curvature metrics. This builds on conjectures of Gromov and on work of Llarull (and others).
Main result: A map f: X -> S^n from a compact Riemannian spin manifold (X,g) to the standard sphere which is 1-Lipschitz and of non-zero degree and such that the scalar curvature of X is everywhere greater or equal to the one of the sphere is automatically an isometry. Here, it is sufficient that the metric is only a Lipschitz metric (which distributional scalar curvature) and that the map is only Lipschitz continuous, not necessarily smooth. New ingredients of proof: combination of Dirac operator tools and the theory of quasiconformal maps. Low regularity index theory on manifolds with conical singularities.
Coauthors: Simone Cecchini, Bernhard Hanke, recently also Lukas Schönlinner
Richard Schoen: tba
Felix Schulze: tba
Carlo Sinestrari: tba
Daniel Stern: New minimal surfaces in \(S^3\) and \(B^3\) via eigenvalue optimization
I'll describe joint work with Karpukhin, Kusner, and McGrath, in which we produce many new families of closed minimal surfaces in \(S^3\) and free boundary minimal surfaces in \(B^3\) via constrained optimization problems for Laplace and Steklov eigenvalues on surfaces. Along the way, I'll highlight some new techniques for establishing existence of extremal metrics in more general situations, and point to some stubborn open problems.
Michael Struwe: The prescribed curvature flow on the disc
For given functions \(f\) and \(j\) on the disc \(B\) and its boundary \(\partial B=S^1\), we study the existence of conformal metrics \(g=e^{2u}g_{\R^2}\) with prescribed Gauss curvature \(K_g=f\) and boundary geodesic curvature \(k_g=j\). Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz in 2018, we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work in 2005 on the prescribed curvature problem on the sphere, we are able to exhibit a \(2\)-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz by degree-theory.
Brian White: Translating Annuli for Mean Curvature Flow
I will discuss suprising new examples of complete minimal annuli that translate under mean curvature flow.
Zhizhang Xie: On Gromov’s dihedral extremality/rigidity conjecture of scalar curvature
In this talk, I will present my joint work with Jinmin Wang and Guoliang Yu on a new index theorem for manifolds with singularities (such as manifolds with corners and more generally for manifolds with polyhedral type boundary). As an application, we obtained a positive solution to Gromov’s dihedral extremality/rigidity conjecture. This conjecture concerns comparisons of scalar curvature, mean curvature and dihedral angles for compact manifolds with polyhedral type boundary, and has very interesting implications in geometry and mathematical physics. Further developments of this new index theorem have led us to a positive solution of Gromov's flat corner domination conjecture. As a consequence, we answered positively a long standing conjecture in discrete geometry - the Stoker conjecture.
Shing-Tung Yau: tba