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After establishing some new global facts (like a measure theoretic structure theorem and a new invariant characterization of Sobolev functions) 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 \(\mathsf {L}^1\)-functions to a class of Riemannian manifolds with possibly unbounded from below Ricci curvature.

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