Li-Yau gradient estimate for compact manifolds with negative part of Ricci curvature in the Kato class

TL;DR

This paper establishes a Li-Yau gradient estimate for heat kernels on compact manifolds where the negative Ricci curvature part is in the Kato class, leading to bounds on topological invariants.

Contribution

It introduces a new heat kernel estimate under Kato class conditions on Ricci curvature and derives topological bounds from this estimate.

Findings

Heat kernel satisfies a Li-Yau type estimate under Kato class Ricci curvature

Bounds on the first Betti number depend only on the Kato constant

Provides new links between curvature conditions and topological invariants

Abstract

This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat kernel bound, we show that the obtained heat kernel estimate leads to bounds on the first Betti number only depending on the Kato constant.