Evolution of Contractions between Non-Compact Manifolds

TL;DR

This paper studies the long-term behavior of mean curvature flow for graphs of length-decreasing maps from Euclidean space to negatively curved manifolds, proving existence, preservation of the graph property, and decay estimates for derivatives.

Contribution

It establishes the global existence and regularity decay estimates for mean curvature flow of length-decreasing maps into negatively curved manifolds.

Findings

Flow exists for all time

The graph property is preserved along the flow

Derivatives of the evolving map decay uniformly

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.