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

TL;DR

This paper establishes that the boundedness of the $L^{3/2}$-norm of scalar curvature determines the extendability of Ricci flow on certain 3-manifolds and characterizes the topology of manifolds with bounded curvature norm.

Contribution

It proves that the $L^{3/2}$-norm of scalar curvature precisely controls Ricci flow extension and classifies manifolds with finite fundamental group and bounded curvature norm.

Findings

Ricci flow extends iff $||R(t)||_{L^{3/2}}$ is bounded

Manifolds with finite fundamental group and bounded $L^{3/2}$-norm are spherical space-forms

Bounded $L^{3/2}$-norm obstructs Ricci flow extension on certain 3-manifolds

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 $0 \leq t < T$ . 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.