Abstract
A surface Σ is a graph in ℝ4 if there is a unit constant 2-form ω on ℝ4 such that <e 1∧e 2, ω≥v 0>0 where {e 1, e 2} is an orthonormal frame on Σ. We prove that, if \( \vartheta _{0} \geqslant \frac{1} {{{\sqrt 2 }}} \) on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M 1×M 2 where M 1 and M 2 are Riemann surfaces, if <e 1∧e 2, ω 1>≥v 0>0 where ω 1 is a Kähler form on M 1. We prove that, if M is a Kähler-Einstein surface with scalar curvature R,\( \vartheta _{0} \geqslant \frac{1} {{{\sqrt 2 }}} \) on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Huisken G., Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom., 1984, 20:237–266
Huisken G., Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math., 1986, 84:463–480
Ecker K., Huisken G., Mean curvature evolution of entire graphs, Ann. Math., 1989, 130:453–471
Altschuler S. J., Singularities of the curve shrinking flow for space curves, J. Diff. Geom., 1991, 34:491–514
Altschuler S. J., Grayson M. A., Shortening space curves and flow through singularities, J. Diff. Geom., 1992, 35:283–298
Huisken G., Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 1990, 31:285–299
Chen J. Y., Li J. Y., Mean curvature flow of surfaces in 4-manifolds, Adv. In Math., (to appear)
Chen J. Y., Tian G., Moving symplectic curves in Kähler-Einstein surfaces, Acta Math. Sinica, English Series, 2000, 16(4):541–548
Osserman R., A Survey of Minimal Surfaces, Van Nostrand, New York, 1969
Lawson H. B., Jr., Osserman R., Non-existence, non-uniqueness andirregu larity of solutions to the minimal surface system, Acta Math., 1977, 139:1–17
Brakke K., The motion of a surface by its mean curvature, Princeton:Princeton University Press, 1978
Spivak M., A Comprehensive Introduction to Differential Geometry, Volume 4, Second Edition, Publish or Perish, Inc. Berkeley, 1979
Author information
Authors and Affiliations
Corresponding author
Additional information
The research is supportedin part by a Sloan fellowship and an NSERC grant for Chen, by a grant from NSF of China for Li, by a grant from NSF of USA for Tian.
Rights and permissions
About this article
Cite this article
Chen, J.Y., Li, J.Y. & Tian, G. Two-Dimensional Graphs Moving by Mean Curvature Flow. Acta Math Sinica 18, 209–224 (2002). https://doi.org/10.1007/s101140200163
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s101140200163