Mean curvature flow of surfaces in Einstein four-manifolds

Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10072

Journal of Differential Geometry, 57, (2), 301-338, 2001
Let  $\sum $ be a compact oriented surface immersed in a four dimensional K¨ahler-Einsteinmanifold $(M, ω)$. We consider the evolution of $\sum$ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When $M$ has two parallel K¨ahler forms $\omega " $ and $\omega ''$ that determine different orientations and  $\sum$ is symplectic with respect to both $\omega " $ and $\omega ''$, we prove the mean curvature flow of $\sum$  exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.
Mean curvature flow, Einstein four-manifolds,
[ Download ] [ 2016-08-22 23:31:27 uploaded by mutaowang ] [ 792 downloads ] [ 0 comments ] [ Cited by 70 ]
@inproceedings{mu-tao2001mean,
  title={Mean curvature flow of surfaces in Einstein four-manifolds},
  author={Mu-Tao Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822233127267992389},
  booktitle={Journal of Differential Geometry},
  volume={57},
  number={2},
  pages={301-338},
  year={2001},
}
Mu-Tao Wang. Mean curvature flow of surfaces in Einstein four-manifolds. 2001. Vol. 57. In Journal of Differential Geometry. pp.301-338. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160822233127267992389.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved