The $L^{3/2}$-norm of the scalar curvature under the Ricci flow on a 3-manifold

Details The $L^{3/2}$-norm of the scalar curvature under the Ricci flow on a 3-manifold The $L^{3/2}$-norm of the scalar curvature under the Ricci flow on a 3-manifold

2011-01-04

arxiv.org/abs/1101.0637v2

Mathematics

Differential Geometry

Revised after the useful comments from R. Haslhofer and H. Li

Scientific paper

Assume $M$ is a closed 3-manifold whose universal covering is not $S^3$. We show that the obstruction to extend the Ricci flow is the boundedness $L^{3/2}$-norm of the scalar curvature $R(t)$, i.e, the Ricci flow can be extended over time $T$ if and only if the $||R(t)||_{L^{3/2}}$ is uniformly bounded for $0 \leq t < T$ . On the other hand, if the fundamental group of $M$ is finite and the $||R(t)||_{L^{\3/2}}$ is bounded for all time under the Ricci flow, then $M$ is diffeomorphic to a 3-dimensional spherical space-form.

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