Heat kernel upper bound on Riemannian manifolds with locally uniform Ricci curvature integral bounds

TL;DR

This paper establishes Gaussian upper bounds for the heat kernel on Riemannian manifolds with locally uniform integral bounds on negative Ricci curvature, linking geometric conditions to analytic and topological properties.

Contribution

It introduces new conditions based on integral Ricci bounds that ensure heat kernel estimates and topological bounds, extending previous results to more general geometric settings.

Findings

Heat kernel admits Gaussian upper bounds under integral Ricci bounds

Function spaces are in the Kato class with these geometric assumptions

Bounds on the first Betti number are derived

Abstract

This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature the heat kernel admits a Gaussian upper bound for small times. This provides general assumptions on the geometry of a manifold such that certain function spaces are in the Kato class. Additionally, the results imply bounds on the first Betti number.