Collapsing in the $L^2$ curvature flow

Details Collapsing in the $L^2$ curvature flow Collapsing in the $L^2$ curvature flow

2012-01-05

arxiv.org/abs/1201.1266v1

Mathematics

Differential Geometry

Scientific paper

We show some results on collapsing phenomena for the $L^2$ curvature flow, which exhibit a stark difference in the behavior of the flow in the subcritical dimensions $n = 2, 3$ and supercritical dimensions $n \geq 5$. First we show long time existence and convergence of the flow for SO(3)-invariant initial data on $S^3$, as well as an optimal long time existence and convergence statement for three-manifolds with initial $L^2$ norm of curvature chosen small with respect to the diameter and volume. 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.

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