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

Abstract

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 $0\le t<T$ . 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.

arXiv commentRevised after the useful comments from R. Haslhofer and H. Li

References in corpus (2)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds