MATH 667-TOPICS IN DIFFERENTIAL GEOMETRY-FALL 2013
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