First-order heat content asymptotics on ${\sf RCD}(K,N)$ spaces

TL;DR

This paper establishes first-order asymptotics for heat content in ${ m RCD}(K,N)$ spaces under a regularity condition on the boundary, linking geometric boundary properties to heat behavior.

Contribution

It introduces and analyzes the measured interior geodesic condition, connecting boundary regularity with heat content asymptotics in ${ m RCD}(K,N)$ spaces.

Findings

Proves first-order heat content asymptotics in ${ m RCD}(K,N)$ spaces.

Relates boundary regularity condition to geometric properties of the space.

Provides a detailed study of the boundary regularity condition and its implications.

Abstract

In this paper, we prove first-order asymptotics on a bounded open set of the heat content when the ambient space is an ${\sf RCD}(K,N)$ space, under a regularity condition for the boundary that we call measured interior geodesic condition of size $\epsilon$. We carefully study such a condition, relating it to the properties of the disintegration of the signed distance function from $\partial \Omega$ studied by Cavalletti and Mondino.