Evolution of contractions by mean curvature flow

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 $${\sec }_M>-\sigma \text{and}{\text{Ric}}_M\ge (m-1)\sigma \ge (m-1){\sec }_N\ge -\mu ,$$ where $\sigma$, $\mu$ are positive constants, we show that the mean curvature flow provides a smooth homotopy of $f$ into a constant map.