Evolution of Contractions between Non-Compact Manifolds

Let $N$ be a complete manifold with bounded geometry, such that $\sec_N\le -\sigma < 0$forsomepositiveconstant$\sigma$. We investigate the mean curvature flow of the graphs of smooth length-decreasing maps $f:{\mathbb{R}}^m\to N$. In this case, the solution exists for all times and the evolving submanifold stays the graph of a length-decreasing map $f_t$. We further prove uniform decay estimates for all derivatives of order $\ge 2$ of $f_t$ along the flow.