Mean curvature flow of area decreasing maps between Riemann surfaces

TL;DR

This paper studies how area decreasing maps between complete Riemann surfaces evolve under mean curvature flow, providing a comprehensive description under specific curvature conditions.

Contribution

It offers a complete analysis of the mean curvature flow for area decreasing maps between Riemann surfaces with bounded geometry, under certain curvature constraints.

Findings

Detailed evolution description of area decreasing maps under mean curvature flow.

Conditions ensuring the flow's well-behaved evolution.

Insights into curvature effects on the flow dynamics.

Abstract

In this article we give a complete description of the evolution of an area decreasing map $f:M\to N$ induced by its mean curvature in the situation where $M$ and $N$ are complete Riemann surfaces with bounded geometry, $M$ being compact, for which their sectional curvatures $\sigma_M$, $\sigma_N$ satisfy $\min\sigma_M\ge\sup\sigma_N$.