The heat flow on glued spaces with varying dimension

TL;DR

This paper introduces glued manifolds with varying dimensions, studies their heat flow properties, and explores the convergence of a nonlocal perimeter functional derived from heat flow to the classical perimeter.

Contribution

It defines a new class of glued manifolds with varying dimensions, analyzes the ergodicity and irreducibility of heat flow on them, and connects heat-based functionals to classical perimeter concepts.

Findings

Heat flow can be ergodic and irreducible under certain conditions.

A nonlocal perimeter functional, the heat excess, is constructed from the heat flow.

Connections are made between heat kernel properties and perimeter functional convergence.

Abstract

In this paper, we introduce a new concept of glued manifolds and investigate under which conditions the canonical heat flow on these glued manifolds is ergodic and irreducible. Glued manifolds are metric spaces consisting of manifolds of varying dimension connected by a weakly doubling measure. This can be seen as a condition on the jump in dimension. From another perspective, this construction also defines the Brownian motion on these glued spaces. Using the heat flow, we construct a nonlocal perimeter functional, the heat excess, to raise the question of its ${\Gamma}$-convergence to the standard perimeter functional. In this context, we connect our work to the previous work on the convergence of perimeter functionals, approximations, and existence of heat kernels, as well as short-time expansions of Brownian motion.