*Published Paper*

**Inserted:** 18 nov 2005

**Last Updated:** 30 mar 2010

**Journal:** J. Reine Angew. Math.

**Volume:** 613

**Year:** 2007

**Abstract:**

Let $M$ be a Riemannian manifold 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 $(T(t))_{t\geq 0}$ be the heat semigroup on $M$. We show that the total variation of the gradient of a function $u\in L^1(M)$ equals the limit of the $L^1$-norm of $\nabla T(t)u$ as $t\to 0$. In particular, this limit is finite if and only if $u$ is a function of bounded variation.

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