Heat kernel bounds and Ricci curvature for Lipschitz manifolds

TL;DR

This paper establishes uniform heat kernel bounds for Lipschitz Riemannian manifolds, enabling analysis of Ricci curvature bounds and function spaces, even in noncomplete or boundary cases.

Contribution

It provides new uniform heat kernel estimates for Lipschitz manifolds and links these bounds to Ricci curvature lower bounds in a synthetic framework.

Findings

Uniform upper bounds on heat kernels decoupled in space and time

Identification of weighted Lebesgue spaces as subsets of the Kato class

Sufficient conditions for manifolds to admit measure-valued Ricci curvature bounds

Abstract

Given any $d$-dimensional Lipschitz Riemannian manifold $(M,g)$ with heat kernel $\mathsf{p}$, we establish uniform upper bounds on $\mathsf{p}$ which can always be decoupled in space and time. More precisely, we prove the existence of a constant $C>0$ and a bounded Lipschitz function $R\colon M \to (0,\infty)$ such that for every $x\in M$ and every $t>0$, \begin{align*} \sup_{y\in M} \mathsf{p}(t,x,y) \leq C\min\{t, R^2(x)\}^{-d/2}. \end{align*} This allows us to identify suitable weighted Lebesgue spaces w.r.t. the given volume measure as subsets of the Kato class induced by $(M,g)$. In the case $\partial M \neq \emptyset$, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on $\partial M$. We use these insights to give sufficient conditions for a possibly noncomplete Lipschitz Riemannian manifold to be tamed, i.e. to admit a measure-valued lower bound on the Ricci curvature, formulated in a synthetic sense.