Mean curvature flow of surface in 4-manifolds
type: Item
instance of: scholarly article
title: Mean curvature flow of surface in 4-manifolds
zbMATH Open document ID: 1002.53046
author: Jingyi Chen
author: Jiayu Li
published in: Advances in Mathematics
publication date: 2002-07-18T00:00:00Z
reviewed by: Christine Guenther
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1703324
zbMATH Keywords: mean curvature flow
zbMATH Keywords: Kähler-Einstein surfaces
zbMATH Keywords: Kähler angle
zbMATH Keywords: holomorphic curves
MaRDI profile type: Publication
full work available at URL: aima.2001.2008
OpenAlex ID: W1991137206
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Wikidata QID: Q125895843
DOI: 10.1006/AIMA.2001.2008
review text: Recall that the mean curvature flow for an embedded surface \(F_{0 }:\Sigma \rightarrow M\) is described by a one-parameter family of smooth maps \(F_{t }\) that satisfies \({d \over dt} F(x,t)=H(x,t), \;F(x,0)=F_{0}(x),\) where \(H(x,t)\) is the mean curvature vector of \(\Sigma_{t} = F_{t} (\Sigma)\) at \(F(x,t)\) in \(M\). A solution has a Type I singularity at a time \(T>0\) if \(\max_{\Sigma_{t}}|A|^{2}\) becomes unbounded, and NEWLINE\[NEWLINE\limsup_{t\rightarrow T}(T-t) \max_{\Sigma_{t}}|A|^{2} \leq C,NEWLINE\]NEWLINE where \(A\) is the second fundamental form. The Kähler angle of the initial surface \(\Sigma_{0}\) is determined by the contraction of the Kähler form of \(M\) with the area element of \(\Sigma_{0}\). NEWLINENEWLINENEWLINEThe authors study the evolution under mean curvature flow of compact real 2-dimensional symplectic surfaces embedded in Kähler-Einstein surfaces. They prove the following: NEWLINENEWLINENEWLINETheorem: Consider a compact real 2-dimensional symplectic surface, which is embedded in a Kähler-Einstein surface with nonnegative scalar curvature. If the cosine of the Kähler angle of the initial surface is greater than 0, then the solution of the mean curvature flow does not exhibit a Type I singularity at any time \(T>0\). NEWLINENEWLINENEWLINEIf there exists a global solution, then the surface converges to holomorphic curves. NEWLINENEWLINENEWLINEThe authors include the evolution equations of the second fundamental form and mean curvature vector for mean curvature flow in arbitrary codimension. The theorem is then proved by contradiction, using a dilation about singularities argument, and a monotonicity formula for NEWLINE\[NEWLINE\int_{\Sigma_{t}}{1 \over cos(\alpha)}\rho(F,t)\varphi d\mu_{t },NEWLINE\]NEWLINE where \(\alpha\) is the Kähler angle, \(\rho\) is the backward heat kernel, and \(\varphi\) is a cutoff function.
description: scientific article; zbMATH DE number 1703324