Parabolic Frequency for the Mean Curvature Flow

Abstract

Abstract This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows (MCFs) and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on MCF shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.