Mean curvature flow of area decreasing maps between Riemann surfaces

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 $\ddot{I}{ƒ}_M$, $\ddot{I}{ƒ}_N$ satisfy $\min \ddot{I}{ƒ}_M\ge \sup \ddot{I}{ƒ}_N$.