Stability of heat kernel bounds under pointed Gromov--Hausdorff convergence

TL;DR

This paper proves that heat kernel bounds remain stable under pointed Gromov--Hausdorff convergence by constructing Dirichlet forms and demonstrating Mosco convergence of energy forms.

Contribution

It introduces a method to establish heat kernel stability and Mosco convergence on limit spaces in the Gromov--Hausdorff setting.

Findings

Heat kernel estimates are stable under pointed Gromov--Hausdorff convergence.

Constructs a conservative, strongly local Dirichlet form on limit spaces.

Shows Mosco convergence of energy forms along subsequences.

Abstract

We construct a conservative and strongly local regular symmetric Dirichlet form on the pointed Gromov--Hausdorff limit space and demonstrate the stability of heat kernel estimates under this convergence. Furthermore, we establish the Mosco convergence of the associated energy forms along a subsequence.