## Abstract

A surface ∑ is a graph in ℝ^{4} if there is a unit constant 2-form ω on ℝ^{4} such that (e_{1} ∧ a_{2}, ω) ≥ v_{0} > 0 where {e_{1},a_{2}} is an orthonormal frame on ∑. We prove that, if v_{0} ≥ 1/2√ 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 ∑ is a graph in M_{1} × M_{2} where M_{1} and M_{2} are Riemann surfaces, if (e_{1} ∧ e_{2},ω1) ≥ 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, V_{0} ≥ 1/√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.

Original language | English (US) |
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Pages (from-to) | 209-224 |

Number of pages | 16 |

Journal | Acta Mathematica Sinica, English Series |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2002 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

## Keywords

- 2-dimensional graphs in ℝ
- Mean curvature flow
- Self-similar solution