The topic for this semester will be: Mean Curvature Flow and Inverse Mean Curvature Flow.

- smooth solutions
- weak solutions: level set solutions, geometric measure theory
- formation of singularities, partial regularity, flows with surgery
- singularities in the mean-convex case
- applications: geometric inequalities, isoperimetric profile, GR-type quantities for AF manifolds

Bibliography for the course

8/21: Overview: MCF for hypersurfaces and graphs, mean curvature motion

Evolution of Curves and Surfaces by Mean Curvature

(survey by Brian White, ICM 2002, Beijing)

Introduction to MCF

Evolution of the geometry under MCF

(Notes I wrote in 2008)

8/23 F MCF: short-time existence, reparametrization of MCM

8/26 M max principle: sphere inclusion/avoidance for MCF

8/28 W max principle: hyperboloid argument, intrinsic heat eqn for extrinsic functions

8/30 F Convex MCF: evolution of curvature

9/4 W Convex MCF: Simons identity, preservation of convexity conditions, Hamilton's MP

9/6 F Convex MCF: tracefree 2nd FF estimates

9/9 M Convex MCF: evolution of f_sigma

9/11 W Convex/mean-convex MCF: intro to gradient estimate

9/13 F Gradient estimate for convex MCF

9/16 M Application: mean curv ratio tends to one/ higher-order derivatives

9/18 W Higher-order derivatives, continuation criterion

9/20 F first order quantities: support function, star-shaped domains