Heat kernel estimate for the Laplace-Beltrami operator under Bakry-\'Emery Ricci curvature condition and applications

TL;DR

This paper derives Gaussian upper bounds for the heat kernel on Riemannian manifolds with Bakry-Émery Ricci curvature bounds, and applies these results to Liouville properties and eigenvalue estimates.

Contribution

It provides new heat kernel estimates under Bakry-Émery curvature conditions and explores their implications for harmonic functions and spectral bounds.

Findings

Established Gaussian upper bounds for the heat kernel.

Proved an L^1-Liouville property for non-negative subharmonic functions.

Derived lower bounds for eigenvalues and upper bounds for the bottom spectrum.

Abstract

We establish a Gaussian upper bound of the heat kernel for the Laplace-Beltrami operator on complete Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below. As applications, we first prove an L^1-Liouville property for non-negative subharmonic functions when the potential function of the Bakry-\'Emery Ricci curvature tensor is of at most quadratic growth. Then we derive lower bounds of the eigenvalues of the Laplace-Beltrami operator on closed manifolds. An upper bound of the bottom spectrum is also obtained.