Evolution of area-decreasing maps between two-dimensional Euclidean spaces

TL;DR

This paper studies the mean curvature flow of area-decreasing maps between two-dimensional Euclidean spaces, proving long-time existence and decay estimates for the evolving graphs.

Contribution

It establishes the long-term existence and decay properties of the mean curvature flow for area-decreasing maps between 2D Euclidean spaces, a novel extension in geometric analysis.

Findings

Solutions exist for all time under area-decreasing condition

Uniform decay estimates for mean curvature vector

Higher-order derivatives of the map decay over time

Abstract

We consider the mean curvature flow of the graph of a smooth map $f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map $f_t$. Further, we prove uniform decay estimates for the mean curvature vector of the graph and all higher-order derivatives of the corresponding map $f_t$.