Singularities of symplectic and Lagrangian mean curvature flows

Abstract

In this paper, we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a Käahler-Einstein surface. We prove that the blow-up flow Σ_s ^∞ at a singular point (X₀, T₀) of a symplectic mean curvature flow Σ_t or of a Lagrangian mean curvature flow Σ_t is a nontrivial minimal surface in ℝ⁴, if Σ_−∞ ^∞ is connected.

Keywords

Symplectic surface / holomorphic curve / Lagrangian surface / minimal Lagrangian surface / mean curvature flow

References

[1] Chen J., Li J. Mean curvature flow of surface in 4-manifolds. Adv Math, 2001, 163: 287-309.

[2] Chen J., Li J. Singularity of mean curvature flow of Lagrangian submanifolds. Invent Math, 2004, 156: 25-51.

[3] Chen J., Tian G. Minimal surfaces in Riemannian 4-manifolds. Geom Funct Anal, 1997, 7: 873-916.

[4] Cheng S. Y., Yau S. T. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math, 1975, 28(3): 333-354.

[5] Chern S. S., Wolfson J. Minimal surfaces by moving frames. Amer J Math, 1983, 105: 59-83.

[6] Ecker K., Huisken G. Mean curvature evolution of entire graphs. Ann Math, 1989, 130: 453-471.

[7] Fujimoto H. Osserman R. Nevanlinna theory and minimal surfaces. Geometry V, Minimal Surfaces, 1997, Berlin: Springer-Verlag, 95-151.

[8] Han X., Li J. The mean curvature flow approach to the symplectic isotopy problem. IMRN, 2005, 26: 1611-1620.

[9] Hoffman D. A., Osserman R. The Geometry of the Generalized Gauss Map. Memoirs of AMS, No 236, 1980, Providence: Amer Math Soc.

[10] Huisken G. Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent Math, 1986, 84: 463-480.

[11] Huisken G. Asymptotic behavior for singularities of the mean curvature flow. J Diff Geom, 1990, 31: 285-299.

[12] Huisken G., Sinestrari C. Mean curvature flow singularities for mean convex surfaces. Calc Var PDE, 1999, 8: 1-14.

[13] Huisken G., Sinestrari C. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math, 1999, 183: 45-70.

[14] Neves A. Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent Math, 2007, 168(3): 449-484.

[15] Simon L. Lectures on Geometric Measure Theory. Proc Center Math Anal, 1983, Canberra: Australian National Univ Press.

[16] Smoczyk K. Harnack inequality for the Lagrangian mean curvature flow. Calc Var PDE, 1999, 8: 247-258.

[17] Smoczyk K. Der Lagrangesche mittlere Kruemmungsfluss. Univ Leipzig (Habil - Schr),102 S. 2000

[18] Smoczyk K. Angle theorems for the Lagrangian mean curvature flow. Math Z, 2002, 240: 849-883.

[19] Thomas R, Yau S T. Special Lagrangians, stable bundles and mean curvature flow. math.DG/0104197, 2001

[20] Wang M. -T. Mean curvature flow of surfaces in Einstein four manifolds. J Diff Geom, 2001, 57: 301-338.

[21] White B. The nature of singularities in mean curvature flow of mean-convex sets. J Amer Math Soc, 2003, 16: 123-138.