Remarks on the curvature behavior at the first singular time of the Ricci flow

Abstract

@@ t gi jD 2Ri j; t2T0; T/; on a smooth, compact n-dimensional Riemannian manifold M. If the flow has uniformly bounded scalar curvature and develops Type I singularities at T , we show that suitable blow-ups of the evolving metrics converge in the pointed Cheeger‐Gromov sense to a Gaussian shrinker by using Perelman’s W-functional. If the flow has uniformly bounded scalar curvature and develops Type II singularities at T , we show that suitable scalings of the potential functions in Perelman’s entropy functional converge to a positive constant on a complete, Ricci flat manifold. We also show that if the scalar curvature is uniformly bounded along the flow in certain integral sense then the flow either develops a Type II singularity at T or it can be smoothly extended past time T .