Concentration-Compactness and Finite-Time Singularities for Chen's Flow

J. Math. Sci. Univ. Tokyo
Vol. 26 (2019), No. 1, Page 55-139.

Bernard, Yann; Wheeler, Glen; Wheeler, Valentina-Mira
Concentration-Compactness and Finite-Time Singularities for Chen's Flow
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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.

Keywords: Curvature flow, global differential geometry, fourth order, geometric analysis, biharmonic, Chen conjecture.

Mathematics Subject Classification (2010): 53C44, 58J35.
Mathematical Reviews Number: MR3929519

Received: 2018-03-01