Li-Yau gradient bounds on compact manifolds under nearly optimal curvature conditions

TL;DR

This paper establishes Li-Yau gradient bounds for the heat equation on manifolds with nearly optimal curvature conditions, extending previous results to cases with unbounded Ricci curvature and under Ricci flow.

Contribution

It introduces new Li-Yau gradient bounds under weaker curvature conditions, including unbounded Ricci curvature and scalar curvature bounds during Ricci flow.

Findings

Gradient bounds hold under $|Ric^-| \, \in L^p$ for $p>n/2$

First Li-Yau bounds allowing unbounded Ricci curvature

Application demonstrating near-optimality of conditions

Abstract

We prove Li-Yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the Ricci flow. In the former case the curvature condition is $|Ric^-| \in L^p$ for some $p>n/2$, or $\sup_\M \int_\M |Ric^-|^2(y)d^{2-n}(x,y)dy<\infty$, where $n$ is the dimension of the manifold. In the later case, one only needs scalar curvature being bounded. We will explain why the conditions are nearly optimal and give an application. The Li-Yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing Ricci curvature not bounded from below.