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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  &gt; 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.

Heat semigroup and functions of bounded variation on Riemannian manifolds