In general it is an open problem how to obtain explicit $C^2$-estimates for graphical submanifolds evolving by its mean curvature. We present some new techniques to get explicit estimates of the mean curvature and in some cases also of the full second fundamental tensor.

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