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 is an element of L-1 (M) equals the limit of the L-1-norm of del T(t)u as t -> 0. In particular, this limit is finite if and only if u is a function of bounded variation.