Parabolic frequency monotonicity on compact manifolds

TL;DR

This paper investigates the behavior of parabolic frequency for heat equation solutions on compact Riemannian manifolds, establishing almost monotonicity under certain curvature conditions and deriving related applications.

Contribution

It generalizes existing monotonicity results to compact manifolds with nonnegative curvature using a generalized matrix Harnack inequality.

Findings

Parabolic frequency is almost increasing on compact manifolds with nonnegative sectional curvature.

A new unique continuation result is obtained.

Monotonicity of a related quantity under Ricci flow is established.

Abstract

This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by L. Ni. The proof is based on a generalization of R. Hamilton's matrix Harnack inequality for small time. As applications, we obtain a unique continuation result. Monotonicity of a new quantity under two-dimensional Ricci flow, closely related to the parabolic frequency functional, is derived as well.