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

TL;DR

This paper studies Chen's flow, a fourth-order curvature flow, proving finite-time extinction of closed submanifolds, characterizing singularities via curvature concentration, and showing smooth convergence to spheres under small initial curvature conditions.

Contribution

It provides the first finite-time extinction results for Chen's flow, characterizes singularity formation through curvature concentration, and establishes conditions for smooth convergence to spheres.

Findings

Closed submanifolds become extinct in finite time.

Singularities are characterized by curvature concentration in $L^n$ for $n=2,4$.

Small initial $L^2$ curvature ensures smooth convergence to a sphere.

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 $L^n$ for intrinsic dimension $n \in \{2,4\}$ and any codimension (a Lifespan Theorem); and (3) for $n = 2$ and in one codimension only, there exists an explicit small constant $\varepsilon_2$ 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.