We study the curvature behavior at the first singular time of a solution to the Ricci flow 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.
|Original language||English (US)|
|Number of pages||21|
|Journal||Pacific Journal of Mathematics|
|State||Published - 2012|
All Science Journal Classification (ASJC) codes
- Ricci flow
- Scalar curvature