# Graduate Student / Postdoc Seminar

#### Mean Curvature Flow of Hypersurfaces

**Speaker:**
Robert Haslhofer

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, February 22, 2013, 1 p.m.

**Synopsis:**

The mean curvature flow evolves hypersurfaces in time: the velocity at each point is given by the mean curvature vector. It is the most natural geometric evolution equation for submanifolds and shares many interesting features with Hamilton's Ricci flow for Riemannian manifolds. In this talk, I will start by explaining and motivating the equation. In particular, I will explain that the mean curvature flow can be viewed as geometric heat equation and thus improves the geometry. To illustrate this, I will describe Huisken's classical result that any convex hypersurface evolves into a round sphere. Finally, I will discuss the formation of singularities in the flow of mean convex hypersurfaces. This is based on fundamental results due to White and Huisken-Sinestrari, and on my recent joint work with Cheeger-Naber and Kleiner.