Curvature contraction of convex hypersurfaces by non-smooth speeds

Glen Wheeler
University of Wollongong
19 May 2014 14:00-15:00, Carslaw Room 829 (AGR)

Abstract

In fully nonlinear curvature flow we typically aim to prove that suitably pinched initial data shrinks to a point in finite time, becoming asymptotically close to a self-similar solution as it does so. Theorems of this type trace back to Huisken?s seminal contribution for the mean curvature flow of convex hypersurfaces. Since then, the efforts of a number of researchers have established similar results for quite broad classes of (typically fully nonlinear) curvature flow with smooth velocity. Our present contribution is an analysis of the case where the speed of the flow, as a function of the eigenvalues of the Weingarten map, is not differentiable. Other requirements on the flow speed are reminiscent of Andrews? early work from 1993/1994. Our main result is that up to necessary modifications, a Huisken-esque result holds.

In this talk I will describe the proof of this result, highlighting the essential new contributions, and mention possible future directions. This work is joint with Ben Andrews (ANU), Andrew Holder (UOW), James McCoy (UOW), Valentina-Mira Wheeler (UOW), and Graham Williams (UOW).

Check also the PDE Seminar page. Enquiries to Daniel Hauer or Daniel Daners.