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.
Categories: math.MG math.AP math.DG