Parabolic frequency for the mean curvature flow

TL;DR

This paper introduces a parabolic frequency concept for heat equation solutions on mean curvature flow shrinkers, proving its monotonicity and deriving backwards uniqueness results, with bounds for more general parabolic equations.

Contribution

It defines a new parabolic frequency for solutions on mean curvature flow shrinkers and proves its monotonicity, leading to backwards uniqueness results.

Findings

Frequency is monotone along mean curvature flow shrinkers.

Backwards uniqueness is established for solutions of the heat equation.

Bounds on frequency derivatives extend to general parabolic equations.

Abstract

This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows 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 mean curvature flow shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.