Deforming area preserving diffeomorphism of surfaces by mean curvature flow

Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10071

Mathematical Research Letters, 8, (5), 651-662., 2001
Let $f : \sum_1 \to \sum_2$ be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of $f$ can be viewed as a Lagrangian submanifold in $ \sum_1 \times \sum_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 $ \sum_1 \times \sum_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 .
Riemann surfaces , diffeomorphism group, mean curvature flow, lagrangian submanifold
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@inproceedings{mu-tao2001deforming,
  title={Deforming area preserving diffeomorphism of surfaces by mean curvature flow},
  author={Mu-Tao Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822232324781699388},
  booktitle={Mathematical Research Letters},
  volume={8},
  number={5},
  pages={651-662.},
  year={2001},
}
Mu-Tao Wang. Deforming area preserving diffeomorphism of surfaces by mean curvature flow. 2001. Vol. 8. In Mathematical Research Letters. pp.651-662.. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822232324781699388.
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