Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow

2001/10/01 by Mu-Tao Wang, Mu‐Tao Wang, Wang, Mu-Tao
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.DG

paper · pdf · doi:10.48550/arxiv.math/0110020

13 pages, to be published in Mathematical Research Letter

arxiv created 2001/10/01 · openalex publication_date 2001/10/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 .

Citations

• A canonical way to deform a Lagrangian submanifold

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