## Journal of Differential Geometry

### Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

Mu-Tao Wang

#### Abstract

Let Σ be a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold (M, w). We consider the evolution of Σ 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 Kähler forms w' and w" that determine different orientations and Σ is symplectic with respect to both w' and w", we prove the mean curvature flow of Σ exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

#### Article information

Source
J. Differential Geom., Volume 57, Number 2 (2001), 301-338.

Dates
First available in Project Euclid: 20 July 2004

https://projecteuclid.org/euclid.jdg/1090348113

Digital Object Identifier
doi:10.4310/jdg/1090348113

Mathematical Reviews number (MathSciNet)
MR1879229

Zentralblatt MATH identifier
1035.53094

#### Citation

Wang, Mu-Tao. Mean Curvature Flow of Surfaces in Einstein Four-Manifolds. J. Differential Geom. 57 (2001), no. 2, 301--338. doi:10.4310/jdg/1090348113. https://projecteuclid.org/euclid.jdg/1090348113