Mathematical Research Letters
Volume 8 (2001)
Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow
Pages: 651 – 661
Let $f:\Sigma_1\rightarrow \Sigma_2 $ be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of $f$ can be viewed as a Lagrangian submanifold in $\Sigma_1\times \Sigma_2$. This article discusses a canonical way to deform $f$ along area preserving diffeomorphisms. This deformation process is realized through the mean curvature flow of the graph of $f$ in $\Sigma_1\times \Sigma_2$. It is proved that the flow exists for all time and the map converges to a canonical map. In particular, this gives a new proof of the classical topological results that $O(3)$ is a deformation retract of the diffeomorphism group of $S^2$ and the mapping class group of a Riemman surface of positive genus is a deformation retract of the diffeomorphism group.