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Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow

Mu-Tao Wang
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Abstract

Let f:Σ_1 --> Σ_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 Σ_1\times Σ_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 Σ_1\times Σ_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 .

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Mu-Tao Wang. 2001-10-01. Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow. https://arxiv.org/abs/math/0110020

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