Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow

Let f:\Sigma_1 --> \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 .