Notre Dame Geometric Analysis Seminar

Anyone is welcome to attend. We meet Thursdays at 11am EST, unless otherwise specified. For the foreseeable future our seminar is held on zoom. Email Nick Edelen (nedelen at nd dot edu) for the zoom password.

Spring 2021

Fall 2020

September 10: 2pm-3pm, Felix Klein seminar

Jiewon Park, MIT

September 17: 11am-12pm, zoom link

Felix Schulze, University of Warwick
Mean curvature flow with generic initial data
We show that the mean curvature flow of generic closed surfaces in R^3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R^4 is smooth until it disappears in a round point. 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 shrinking solitons. This is joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.

September 24: 11am-12pm, zoom link

Chao Li, Princeton
Generalized soap bubbles and the topology of manifolds with positive scalar curvature
It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. I will present some recent progress on this question, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov), as well as complete metrics of positive scalar curvature on an arbitrary manifold connect sum with a torus. Applications include a Schoen-Yau Liouville theorem for all locally conformally flat manifolds. The proof of these results are based on analyzing generalized soap bubbles - surfaces that are stable solutions to the prescribed mean curvature problem. This talk is based on joint work with O. Chodosh.

October 15: 11am-12pm, zoom link

Samuel Perez-Ayala, University of Notre Dame
Extremal Metrics for the Conformal Laplacian
In this talk I will discuss the existence and properties of extremal metrics for the spectrum of the Conformal Laplacian. As I will explain, depending on various conformally invariant quantities, such extremal metrics are associated with either constant scalar curvature metrics (Yamabe metrics), nodal solutions to known PDEs or Harmonic maps into spheres. This is joint work with Matt Gursky.

October 22: 12pm-1pm (NOTE TIME), zoom link

Yevgeny Liokumovich, University of Toronto
Generic regularity of min-max minimal hypersurfaces
Minimal hypersurfaces in 8-dimensional Riemannian manifolds may have isolated singularities. However, it follows from the results of R. Hardt, L. Simon and N. Smale that one can perturb away singularities of an area minimizing minimal hypersurface by a small change of the metric. I will talk about a similar problem for min-max minimal hypersurfaces (joint work with Otis Chodosh and Luca Spolaor). We show that for a generic 8-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics there exists a minimal hypersurface with at most one singular point. Our proof uses a construction of optimal nested sweepouts from a joint work with Gregory Chambers.

October 29: 11am-12pm, zoom link

Andrea Mondino, University of Oxford
Optimal transport and quantitative geometric inequalities
The goal of the talk is to discuss a quantitative version of the Levy- Gromov isoperimetric inequality (joint with Cavalletti and Maggi) as well as a quantitative form of Obata's rigidity theorem (joint with Cavalletti and Semola). Given a closed Riemannian manifold with strictly positive Ricci tensor, one estimates the measure of the symmetric difference of a set with a metric ball with the deficit in the Levy- Gromov inequality. The results are obtained via a quantitative analysis based on the localisation method via L1-optimal transport. For simplicity of presentation, the talk will present the results in case of smooth Riemannian manifolds with Ricci Curvature bounded below; moreover it will not require previous knowledge of optimal transport theory.

November 5: 11am-12pm, zoom link

Aaron Tyrrell, University of Notre Dame
Renormalized Area
We will look at some results regarding the renormalised area of minimal submanifolds of Poincare-Einstein manifolds.

November 5: 2pm-3pm (NOTE TIME), zoom link

Zhehui Wang, Beijing International Center for Mathematical Research
A Bernstein Type Theorem for Minimal Graphs over Convex Cones
In this talk, we study a Bernstein type theorem for minimal graphs over convex cones. Namely, any solution of minimal surface equation in convex cones must be linear, if it agrees with a linear function on the boundary. With such a boundary requirement, we do not need dimension restriction for this result. This talk is based on joint work with Nick Edelen.

November 12, 11am-12pm, zoom link

Patrick Heslin, University of Notre Dame
A regularising property for solutions to the 2D Euler Equations in Lagrangian coordinates
I will present a regularising property of the L2 exponential map on the group of volume-preserving diffeomorphisms of T^2 of Sobolev-class H^s, whose geodesics, as Arnold illustrated in 1966, correspond to H^s solutions to the Euler equations of hydrodynamics: \partial_t + \nabla_u u = -\nabla p , div(u) = 0 .

November 19, 2pm-3pm, zoom link

Shih-Kai Chiu, University of Notre Dame
A Liouville type theorem for harmonic 1-forms
We prove a Liouville type theorem for harmonic 1-forms. As an application, we show that a harmonic function with polynomially subquadratic growth on a Ricci-flat Kahler manifold with maximum volume growth must be pluriharmonic.