Evolution of contractions by mean curvature flow

TL;DR

This paper studies how mean curvature flow can deform length decreasing maps between Riemannian manifolds into constant maps under certain curvature conditions, extending understanding of geometric flows in differential geometry.

Contribution

It establishes conditions under which mean curvature flow smoothly homotopes length decreasing maps into constant maps, advancing the theory of geometric flows on manifolds.

Findings

Mean curvature flow deforms maps into constants under curvature bounds

Conditions involve sectional and Ricci curvature constraints

Flow provides smooth homotopies between maps

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