School of Mathematical Sciences, Monash University
Centre for Geometric Analysis, Institute for Mathematics and its Applications, Faculty of Informatics and Engineering, University of Wollongong
抄録
Chen's flow is a fourth-order curvature flow motivated by the spectral decomposition of immersions, a program classically pushed by B.-Y. Chen since the 1970s. In curvature flow terms the flow sits at the critical level of scaling together with the most popular extrinsic fourth-order curvature flow, the Willmore and surface diffusion flows. Unlike them however the famous Chen conjecture indicates that there should be no stationary nonminimal data, and so in particular the flow should drive all closed submanifolds to singularities. We investigate this idea, proving that (1) closed data becomes extinct in finite time in all dimensions and for any codimension; (2) singularities are characterised by concentration of curvature in $L^n$ for intrinsic dimension $n \in \{2,4\}$ and any codimension (a Lifespan Theorem); and (3) for $n=2$ and in any codimension, there exists an explicit $\varepsilon_2$ such that if the $L^2$ norm of the tracefree curvature is initially smaller than $\varepsilon_2$, the flow remains smooth until it shrinks to a point, and that the blowup of that point is an embedded smooth round sphere.
雑誌名
Journal of Mathematical Sciences The University of Tokyo
巻
26
号
1
ページ
55 - 139
発行年
2019-03-22
ISSN
13405705
書誌レコードID
AA11021653
Mathmatical Subject Classification
53C44(MSC2010)
58J35(MSC2010)
出版者
Graduate School of Mathematical Sciences, The University of Tokyo
原稿受領日
2018-03-01
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Concentration-Compactness and Finite-Time Singularities for Chen's Flow
jms260103.pdf
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