Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

By Mu-Tao Wang. 2001

183 citations recorded by the source corpus.

Let Σ be a compact oriented surface immersed in a four dimensional KahlerEinstein manifold (M,ω). 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 Kahler forms ω′ and ω′′ that determine different orientations and Σ is symplectic with respect to both ω′ and ω′′, we prove the mean curvature flow of Σ exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cba621bc21d1e68de25739c88c33aa309e5b93ef

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Knut Smoczyk
2. Mao-Pei Tsui
3. Jingyi Chen
4. Jiayu Li
5. Mu-Tao Wang
6. C. Baker
7. D. Joyce
8. Y. Xin

In Geometry and Topology

1. Knut Smoczyk
2. Jingyi Chen
3. Mao-Pei Tsui
4. Jiayu Li
5. Mu-Tao Wang
6. C. Baker
7. D. Joyce
8. Y. Xin

Citations in the corpus

• Mean curvature flows and isotopy of maps between spheres (2003, background): “The calculation indeed already appears in [14].”
• A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension (2011, unknown): “[14],[13],[12],[17]) 0 = 1 2 ∆|A|(2) = ∑”
• Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, background): “of the results of Wang, Chen and Li in [17, 86] w.”
• On graphic Bernstein type results in higher codimension (2002, background): “It is shown in [12] that, under this assumption, the maximum principle not only works in the elliptic case but also in the parabolic case.”
• Singularity of mean curvature flow of Lagrangian submanifolds (2003, background): “A straightforward calculation (cf. [CL1], [H1], [ Wa ]) shows”
• Self-similar solutions and translating solitons for Lagrangian mean curvature flow (2008, unknown): “For more general Lagrangian self-similar solutions, it includes the examples of Abresch and Langer [2] when n = 1, the examples of Anciaux [1], which have λ 1 = · · · = λ n = C = 1 and are symmetric under the action of SO(n) on C n , and the examples of Lee and Wang [17, §6], [18], which have w j (s”
• Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension (2001, background): “∂fα ∂t = (gij + hγβ ∂f γ ∂xi ∂fβ ∂xj )−1 ∂2fα ∂xi∂xj”