  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
$\min\sigma_M\ge\sup\sigma_N$.
