The $L^{\frac{3}{2}}$-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 $L^{\frac{3}{2}}$-norm of the scalar curvature R(t), i.e., the Ricci flow can be extended over finite time T if and only if the $\Vert R(t)\Vert_{L^{\frac{3}{2}}}$ is uniformly bounded for 0≤t<T. On the other hand, if the fundamental group of M is finite and the $\Vert R(t)\Vert_{L^{\frac{3}{2}}}$ is bounded for all time under the Ricci flow, then M is diffeomorphic to a 3-dimensional spherical space-form.

Keywords

Ricci flow / Long time existence