Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow

Let f : Σ1 → Σ2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in Σ1 × Σ2. 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 Σ1 × Σ2. 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 S2 and the mapping class group of a Riemman surface of positive genus is a deformation retract of the diffeomorphism group .