Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below

TL;DR

This paper extends the theory of functions with bounded variation to a broad class of noncompact Riemannian manifolds with unbounded Ricci curvature, establishing new measure-theoretic and approximation results.

Contribution

It introduces new global structural results for BV functions on general Riemannian manifolds and extends heat semigroup characterizations to manifolds with unbounded Ricci curvature.

Findings

New measure theoretic structure theorem for BV functions

Approximation results for BV functions on noncompact manifolds

Extension of heat semigroup characterization to manifolds with unbounded Ricci curvature

Abstract

After establishing some new global facts (like a measure theoretic structure theorem and approximation results) about complex-valued functions with bounded variation on arbitrary noncompact Riemannian manifolds, we extend results of Miranda/the second author/Paronetto/Preunkert and of Carbonaro/Mauceri on the heat semigroup characterization of the variation of L^1-functions to a class of Riemannian manifolds with possibly unbounded from below Ricci curvature.