Heat kernel bounds and Ricci curvature for Lipschitz manifolds

2021/11/24 by Braun, Mathias, Rigoni, Chiara
#47D08 #53C23 #58J35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 51F30 #Probability (math.PR) #Secondary: 46E36

paper · doi:10.48550/arxiv.2111.12607

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\colon M → (0,∞) such that for every x∈ M and every t>0, sup_{y∈ M} p(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 ≠ ∅, 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.

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