Collapsing in the L 2 Curvature Flow

We show some results for the L² 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 S³, as well as a long time existence and convergence statement for three-manifolds with initial L² 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 ≥ 5, and show examples of finite time singularities in dimension n ≥ 6 which are collapsed on the scale of curvature.