Heat kernel on smooth metric measure spaces and applications

TL;DR

This paper establishes fundamental heat kernel estimates and inequalities on smooth metric measure spaces with Bakry-Émery curvature bounds, leading to applications in Liouville theorems, uniqueness, and eigenvalue estimates.

Contribution

It derives sharp Gaussian bounds and Harnack inequalities for the $f$-heat kernel on such spaces, extending classical results to the weighted setting.

Findings

Sharp Gaussian upper and lower bounds for the $f$-heat kernel

Harnack inequality for positive solutions of the $f$-heat equation

Applications to Liouville theorems, uniqueness, and eigenvalue estimates

Abstract

We derive a Harnack inequality for positive solutions of the $f$-heat equation and Gaussian upper and lower bounds for the $f$-heat kernel on complete smooth metric measure spaces $(M, g, e^{-f}dv)$ with Bakry-\'Emery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser theory. As applications, we prove an $L^1_f$-Liouville theorem for $f$-subharmonic functions and an $L^1_f$-uniqueness theorem for $f$-heat equations when $f$ has at most linear growth. We also obtain eigenvalues estimates and $f$-Green's function estimates for the $f$-Laplace operator.