A rigidity result for metric measure spaces with Euclidean heat kernel

TL;DR

This paper proves that metric measure spaces with an Euclidean heat kernel are isometric to Euclidean space, offering an alternative proof of a volume rigidity theorem and exploring spherical heat kernels.

Contribution

It establishes a rigidity result linking heat kernel properties to the geometry of the space, providing a new proof of an existing volume theorem.

Findings

Spaces with Euclidean heat kernels are Euclidean spaces

Provides an alternative proof of Colding's volume theorem

Discusses spherical heat kernel cases

Abstract

We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.