Sets of finite perimeter on Riemannian manifolds and stochastic completeness

TL;DR

This paper characterizes the total variation of BV functions on Riemannian manifolds using heat semigroups and provides a counterexample where this characterization fails.

Contribution

It extends heat semigroup characterizations of BV functions to weighted Riemannian manifolds and shows limitations through a specific counterexample.

Findings

Heat semigroup characterizes total variation for BV functions on compact weighted manifolds.

An example where the heat semigroup characterization does not hold for certain sets.

Extension of previous results to more general Riemannian settings.

Abstract

We prove a heat semigroup characterization of the total variation for compactly supported ${\rm BV}$ on arbitrary smooth complete weighted Riemannian manifolds, extending the main result in \cite{GP15}. We then provide an example of a weighted manifold where such equivalence does not hold for a large class of sets of finite perimeter.