Evolution of contractions by mean curvature flow

Abstract

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