Collapsing in the $L^2$ curvature flow

TL;DR

This paper investigates the behavior of the $L^2$ curvature flow, demonstrating long-term existence and convergence in certain cases, and identifying finite-time singularities in higher dimensions.

Contribution

It provides new results on long-time existence, convergence, and singularity formation for the $L^2$ curvature flow across various dimensions and initial conditions.

Findings

Long-time existence and convergence for $SO(3)$-invariant data on $S^3$

Existence and convergence for small $L^2$ curvature norm relative to diameter and volume

Finite-time singularities in dimensions $n \,\geq\, 5$ and collapsed singularities in $n \,\geq\, 6$

Abstract

We show some results for the $L^2$ curvature flow linked by the theme of addressing collapsing phenomena. First we show long time existence and convergence of the flow for $SO(3)$-invariant initial data on $S^3$, as well as a long time existence and convergence statement for three-manifolds with initial $L^2$ norm of curvature chosen small with respect only to the diameter and volume, which are both necessary dependencies for a result of this kind. In the critical dimension $n = 4$ we show a related low-energy convergence statement with an additional hypothesis. Finally we exhibit some finite time singularities in dimension $n \geq 5$, and show examples of finite time singularities in dimension $n \geq 6$ which are collapsed on the scale of curvature.