Gaussian upper bounds for the heat kernel on evolving manifolds
arXiv:2007.07112 · doi:10.1112/jlms.12793
Abstract
In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method which was previously used by Davies in the case of a non-evolving manifold. This result directly implies Gaussian-type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.
arXiv comment20 pages
References in corpus (5)
- The logarithmic Sobolev inequality along the Ricci flow
- Perelman's Entropy Functional at Type I Singularities of the Ricci Flow
- Monotone Volume Formulas for Geometric Flows
- Bounds on the Heat Kernel under the Ricci Flow
- Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons