Evolution of Contractions between Non-Compact Manifolds

Abstract

Let $N$ be a complete manifold with bounded geometry, such that ${\sec }_N\le -\sigma <0$ for some positive constant $\sigma$. 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.

29 pages; comments are welcome! arXiv admin note: text overlap with arXiv:1608.05394, and with arXiv:1602.07595 by other authors