The Extension of the Hk Mean Curvature Flow in Riemannian Manifolds

In this paper, the authors consider a family of smooth immersions Ft ： Mn → Nn＋1 of closed hypersurfaces in Riemannian manifold Nn＋1 with bounded geometry, mov- ing by the Hk mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the Hk mean curvatur...

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Bibliographic Details
Published inChinese annals of mathematics. Serie B Vol. 35; no. 2; pp. 191 - 208
Main Authors Qiu, Hongbing, Ye, Yunhua, Zhu, Anqiang
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2014
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Subjects
Applications of Mathematics
Mathematics
Mathematics and Statistics
光滑浸入
平均曲率流
扩展定理
最高温度
有限时间区间
第二基本形式
香港
黎曼流形
Sobolev type inequality
mean curvature flow
Riemannian manifold
Moser iteration
53C44
53C21
Hk mean curvature flow
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Summary:In this paper, the authors consider a family of smooth immersions Ft ： Mn → Nn＋1 of closed hypersurfaces in Riemannian manifold Nn＋1 with bounded geometry, mov- ing by the Hk mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the Hk mean curvature flow solution with finite total mean curvature on a finite time interval [0, Tmax） can be extended over Tmax. This result gen- eralizes the extension theorems in the paper of Li （see ＂On an extension of the Hk mean curvature flow, Sci. China Math, 55, 2012, 99-118＂）.
Bibliography:In this paper, the authors consider a family of smooth immersions Ft ： Mn → Nn＋1 of closed hypersurfaces in Riemannian manifold Nn＋1 with bounded geometry, mov- ing by the Hk mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the Hk mean curvature flow solution with finite total mean curvature on a finite time interval [0, Tmax） can be extended over Tmax. This result gen- eralizes the extension theorems in the paper of Li （see ＂On an extension of the Hk mean curvature flow, Sci. China Math, 55, 2012, 99-118＂）.
Hk mean curvature flow, Riemannian manifold, Sobolev type inequality,Moser iteration
31-1329/O1
ISSN:0252-9599
1860-6261
DOI:10.1007/s11401-014-0827-y