Collapsing in the $L^2$ curvature flow

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 $n\ge 5$, and show examples of finite time singularities in dimension $n\ge 6$ which are collapsed on the scale of curvature.

to appear in Comm. PDE

References in corpus (4)

• Ricci flow with surgery on three-manifolds
• Fourth order curvature flows and geometric applications
• The gradient flow of the $L^2$ curvature energy near the round sphere
• The long time behavior of fourth-order curvature flows