Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures

TL;DR

This paper investigates the monotonicity properties of parabolic frequency functions under Ricci flow and Ricci-harmonic flow, considering cases with bounded Ricci and Bakry-Émery Ricci curvatures, advancing understanding of heat equation behavior in geometric flows.

Contribution

It introduces new monotonicity results for parabolic frequency in Ricci and Ricci-harmonic flows with bounded curvatures, extending previous work to broader curvature conditions.

Findings

Monotonicity of parabolic frequency for solutions with bounded Bakry-Émery Ricci curvature.

Monotonicity of parabolic frequency for solutions with bounded Ricci curvature.

Extension of frequency monotonicity results to Ricci-harmonic flow.

Abstract

In this paper, we study the monotonicity of parabolic frequency motivated by \cite{frequency on RF} under the Ricci flow and the Ricci-harmonic flow on manifolds. Here we consider two cases: one is the monotonicity of parabolic frequency for the solution of linear heat equation with bounded Bakry-\'{E}mery Ricci curvature, and another case is the monotonicity of parabolic frequency for the solution of heat equation with bounded Ricci curvature.