Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

2001/10/01 by Mu-Tao Wang, Wang, Mu-Tao · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.math/0110019

39 pages, to be published in Journal of Differential Geometry

arxiv created 2001/10/01 · arxiv updated 2009/11/30

Abstract

Let Σbe a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold M. We consider the evolution of Σin 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 ω' and ω'' that determine different orientations andΣis symplectic with respect to both ω' and ω'', we prove the mean curvature flow of Σexists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

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• Rigidity and Classification of Legendrian Self-Shrinkers
• Lagrangian Translating Solitons and Special Lagrangians in ℂ^m with Symmetries

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