Li-Yau gradient bound for collapsing manifolds under integral curvature condition

TL;DR

This paper establishes Li-Yau gradient bounds for heat equation solutions on collapsing manifolds under integral Ricci curvature conditions, extending previous results to include collapsing scenarios with small integral negative Ricci curvature.

Contribution

It proves Li-Yau type gradient bounds for heat solutions on collapsing manifolds with small integral negative Ricci curvature, broadening the scope of previous non-collapsing results.

Findings

Li-Yau gradient bounds hold under small integral Ricci curvature.

Results apply to collapsing manifolds, not just non-collapsed.

Extends prior work to include integral curvature conditions.

Abstract

Let $(\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup_{x\in M}r^2\left(\oint_{B(x,r)}|Ric^-|^p dV\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac{n}{2}$, when $k(p,1)$ is small enough, certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic balls $B(O,r)$ in $\M$ with $0<r\leq 1$. Here the assumption that $k(p,1)$ being small allows the situation where the manifolds is collapsing. Recall that in \cite{ZZ}, certain Li-Yau gradient bounds was also obtained by the authors, assuming that $|Ric^-|\in L^p(\M)$ and the manifold is noncollaped. Therefore, to some extent, the results in this paper and in \cite{ZZ} complete the picture of Li-Yau gradient bound for the heat equation on manifolds with $|Ric^-|$ being $L^p$ integrable, modulo sharpness of constants.