On symplectic mean curvature flows

Abstract

In this paper we review all the main known results about mean curvature flows with initial surfaces symplectic in a Kähler-Einstein surface, including published results and new results obtained recently. We also propose some problems that we think are very interesting.

Keywords

symplectic surface / holomorphic curve / mean curvature flow (MCF)

References

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

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

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

[4] Smoczyk K. Der Lagrangesche mittlere Kruemmungsfluss. Univ Leipzig (HabilSchr), 102 S. 2000

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

[6] Spivak M. A Comprehensive Introduction to Differential Geometry, Vol 4, 1979 2nd Ed. Berkeley: Publish or Perish, Inc.

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

[8] Hamilton R. Monotonicity formulas for parabolic flows. Comm Anal Geom, 1993, 1(1): 127-137.

[9] Struwe M. On the evolution of harmonic maps in higher dimensions. J Diff Geom, 1988, 28: 485-502.

[10] Chen Y., Struwe M. Existence and partial regularity for heat flow for harmonic maps. Math Z, 1989, 201: 83-103.

[11] White B. A local regularity theorem for mean curvature flow. Ann Math, 2005, 161: 1487-1519.

[12] Chen J., Tian G. Moving symplectic curves in Kähler-Einstein surfaces. Acta Math Sin (Engl Ser), 2000, 16(4): 541-548.

[13] Arezzo C. Minimal surfaces and deformations of holomorphic curves in Kähler-Einstein manifolds. Ann Scuola Norm Sup Pisa Cl Sci (4), 2000, 29(2): 473-481.

[14] Chen J., Li J., Tian G. Two-dimensional graphs moving by mean curvature flow. Acta Math Sin (Engl Ser), 2002, 18: 209-224.

[15] Wang M.-T. Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent Math, 2002, 148: 525-543.

[16] Li J., Li Y. Mean curvature flow of graphs in Σ₁ × Σ₂. J Partial Diff Equat, 2003, 16: 273-282.

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

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

[19] Chen J, Li J. Singularities of codimension two mean curvature flow of symplectic surfaces. preprint

[20] Han X, Li J. Singularities of symplectic and Lagrangian mean curvature flows. preprint