  Let $f:M\to N$ be a smooth area decreasing map between two Riemannian
manifolds $(M,\gm)$ and $(N,\gn)$. Under weak and natural assumptions on the
curvatures of $(M,\gm)$ and $(N,\gn)$, we prove that the mean curvature flow
provides a smooth homotopy of $f$ to a constant map.
