Heat semigroup and functions of bounded variation on Riemannian manifolds

Let M be a connected Riemannian manifold without boundary with Ricci curvature bounded from below and such that the volume of the geodesic balls of centre x and fixed radius r > 0 have a volume bounded away from 0 uniformly with respect to x, and let (T(t)) t≧0 be the heat semigroup on M. We show that the total variation of the gradient of a function u ∈ L 1(M) equals the limit of the L 1-norm of ∇T(t)u as t → 0. In particular, this limit is finite if and only if u is a function of bounded variation.