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,
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  title={Mean curvature flow of surfaces in Einstein four-manifolds},
  author={Mu-Tao Wang},
  booktitle={Journal of Differential Geometry},
Mu-Tao Wang. Mean curvature flow of surfaces in Einstein four-manifolds. 2001. Vol. 57. In Journal of Differential Geometry. pp.301-338.
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