Let f:��_1 --> ��_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 ��_1\times ��_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 ��_1\times ��_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 .

13 pages, to be published in Mathematical Research Letter