Two-Dimensional Graphs Moving by Mean Curvature Flow

Jing Yi Chen, Jia Yu Li, Gang Tian

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

A surface ∑ is a graph in ℝ4 if there is a unit constant 2-form ω on ℝ4 such that (e1 ∧ a2, ω) ≥ v0 > 0 where {e1,a2} is an orthonormal frame on ∑. We prove that, if v0 ≥ 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 M1 × M2 where M1 and M2 are Riemann surfaces, if (e1 ∧ e2,ω1) ≥ V0 > 0 where ω1 is a Kähler form on M1. We prove that, if M is a Kähler-Einstein surface with scalar curvature R, V0 ≥ 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 languageEnglish (US)
Pages (from-to)209-224
Number of pages16
JournalActa Mathematica Sinica, English Series
Volume18
Issue number2
DOIs
StatePublished - Apr 2002

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

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

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