Geometric Analysis Seminar

Autumn 2023

1. Oct 11, 2023 Wed, 9:00-10:00 AM, Zoom

Kai Xu, Duke University

On the existence problem of weak inverse mean curvature flow: Inverse mean curvature flow is the flow that evolves a mean-convex hypersurface at the speed of the inverse of its mean curvature. The focus of this talk is a weak version of the inverse mean curvature flow, defined by Huisken and Ilmanen in 2001. We will introduce this weak formulation and explain the behavior of weak solutions. Then we explain how the existence of weak solutions is affected by the geometry of the manifold. Finally, we introduce the speaker's recent theorem, which roughly states that a certain isoperimetric inequality of a manifold implies the existence of weak solutions on it.

2. Oct 18, 2023 Wed, 10:00-11:00 AM, Zoom

Man-Chun Lee, CUHK

Ricci Flow and pinched curvature on non-compact manifolds: In dimension three, it was proved recently by Deruelle-Schulze-Simon, Lott, Lee-Topping that three-manifolds with non-negative pinched Ricci curvature are compact or flat. In this talk, we will discuss its partial generalisation to higher dimension using the Ricci Flow method. This is based on joint work with P. Topping.

3. Dec 26, 2023 Tue, 9:30-10:30 AM, Zoom

Hong Huang, Beijing Normal University

Topological classification of compact manifolds with positive isotropic curvature: The notion of positive isotropic curvature was introduced by Micallef and Moore in 1988. In this talk I’ll first briefly survey some of the previous (before 2019) works of various authors on Riemannian manifolds with positive isotropic curvature. Then I’ll introduce my recent work on topological classification of compact manifolds of dimension $n \geq 12$ with positive isotropic curvature, which uses Ricci flow with surgery on orbifolds and some techniques from differential topology, and is based on Brendle’s curvature pinching estimates.