  We investigate length decreasing maps $f:M\to N$ between Riemannian manifolds
$M$, $N$ of dimensions $m\ge 2$ and $n$, respectively. Assuming that $M$ is
compact and $N$ is complete such that
$$\sec_M>-\sigma\quad\text{and}\quad{\Ric}_M\ge(m-1)\sigma\ge(m-1)\sec_N\ge-\mu,$$
where $\sigma$, $\mu$ are positive constants, we show that the mean curvature
flow provides a smooth homotopy of $f$ into a constant map.
