Heat kernel bounds and Ricci curvature for Lipschitz manifolds

Abstract

Given any d-dimensional Lipschitz Riemannian manifold (M,g) with heat kernel p, we establish uniform upper bounds on 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:M→(0,∞) such that for every x∈M and every t>0, supy∈Mp(t,x,y)≤Cmin{t,R2(x)}−d/2.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 ∂M≠0̸, we also provide an analogous inclusion for Lebesgue spaces w.r.t. the surface measure on ∂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.