Superconvexity of the heat kernel on hyperbolic space with applications to mean curvature flow

Abstract

We prove a conjecture of Bernstein that the superconvexity of the heat kernel on hyperbolic space holds in all dimensions and, hence, there is an analog of Huisken's monotonicity formula for mean curvature flow in hyperbolic space of all dimensions.

A Sharp Isoperimetric Property of the Renormalized Area of a Minimal Surface in Hyperbolic Space

“…In [3], the author introduced a notion of entropy for submanifolds of hyperbolic space analogous to the one introduced by Colding and Minicozzi in [8] for submanifolds of Euclidean space (see also [16]). More precisely, let H 2 (t, p; t 0 , p 0 ) be the heat kernel on H 2 with singularity at p = p 0 at time t = t 0 .…”