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Collapsing in the $L^2$ curvature flow

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 Added by Jeffrey Streets
 Publication date 2012
  fields
and research's language is English
 Authors Jeff Streets




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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.



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