Heat kernel estimates and the essential spectrum on weighted manifolds

TL;DR

This paper establishes heat kernel upper bounds and analyzes the essential spectrum of the drifting Laplacian on weighted manifolds with lower bounds on the Bakry-Émery Ricci tensor, using warped product space comparisons.

Contribution

It introduces a method to estimate heat kernels on weighted manifolds via warped product spaces and applies this to spectral analysis of the drifting Laplacian.

Findings

Heat kernel upper bounds derived from warped product comparisons.

Essential spectrum characterized under Bakry-Émery Ricci curvature bounds.

Applicable to complete noncompact weighted manifolds.

Abstract

We consider a complete noncompact smooth Riemannian manifold $M$ with a weighted measure and the associated drifting Laplacian. We demonstrate that whenever the $q$-Bakry-\'Emery Ricci tensor on $M$ is bounded below, then we can obtain an upper bound estimate for the heat kernel of the drifting Laplacian from the upper bound estimates of the heat kernels of the Laplacians on a family of related warped product spaces. We apply these results to study the essential spectrum of the drifting Laplacian on $M$.