Conformal Perturbation of Heat Kernels with applications

TL;DR

This paper investigates how conformal changes to a Riemannian metric affect heat kernels, providing bounds and gradient estimates for the perturbed heat kernels on smooth manifolds.

Contribution

It derives new upper bounds and gradient estimates for heat kernels under conformal metric perturbations on Riemannian manifolds.

Findings

Established upper bounds for conformally perturbed heat kernels

Derived gradient estimates for the perturbed heat kernels

Applied results to analyze heat kernel behavior under conformal changes

Abstract

Let $(M, g)$ be a smooth n-dimensional Riemannian manifold for $n\ge 2$. Consider the conformal perturbation $\tilde{g}=h g$ where $h$ is a smooth bounded positive function on $M$. Denote by $\tilde{p}_t(x,y)$ the heat kernel of manifolds $(M, \tilde{g})$. In this paper, we derive the upper bounds and gradient estimates of $\tilde{p}_t(x,y)$.