Concentration-Compactness and finite-time singularities for Chen's flow

Yann Bernard, Glen Wheeler, Valentina Mira Wheeler

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Abstract

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 Ln for intrinsic dimension n ϵ {2,4} and any codimension (a Lifespan Theorem); and (3) for n-2 and in any codimension, there exists an explicit e-i such that if the 1? norm of the tracefree curvature is initially smaller than £2, the flow remains smooth until it shrinks to a point, and that the blowup of that point is an embedded smooth round sphere.

Original languageEnglish
Pages (from-to)55-139
Number of pages85
JournalJournal of Mathematical Sciences
Volume26
Issue number1
Publication statusPublished - 1 Jan 2019

Keywords

  • Biharmonic
  • Chen Conjecture
  • Curvature Flow
  • Fourth Order
  • Geometric Analysis
  • Global Differential Geometry

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