摘要
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 finite time T if and only if the||R(t)||L 3/2 is uniformly bounded for 0≤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.
基金
The author would like to express his gratitude to X.Chen who brought this problem to his attention and provided many helpful and stimulating discussions.He is very grateful of V.Apostolov’s detailed suggestions for this paper.He also would like to thank H.Li for discussing and reviewing the paper and R.Haslhofer for useful comments.
