Harnack Inequalities and Ancient Solutions of Hypersurface Flows

Presenter: Dr Paul Bryan, UQ

Harnack inequalities are an important tool in the analysis of elliptic and parabolic PDE. For geometric flows such at the Ricci flow and the Mean Curvature Flow, the so-called Li-Yau-Hamilton-Harnack inequality plays a central role in a number of places, in particular in the study of singularity formation. Ancient solutions of a parabolic PDE are those solutions existing on an interval $(-\infty,T)$, and are closely related to the Harnack inequality in that solitons (a self-similar class of ancient solutions) achieve equality. Solitons are themselves of importance since they model general singularity formation. To date, most work on hypersurface flows has focused on Euclidean space, where a Harnack inequality for general hypersurface flows was obtained by Ben Andrews via a simple computation using the support function and Gauss map parametrisation. I will a describe a variation of this approach that works in arbitrary Riemannian backgrounds, tying together several different aspects of the Harnack inequality. I will also describe how this relates to ancient solutions, and time permitting will give a brief survey of several classification techniques.