Remarks on the curvature behavior at the first singular time of the Ricci flow
This paper discusses the behavior of the Ricci flow at the first singular time of one solution for the Ricci equation NEWLINE\[NEWLINE\frac{\partial}{\partial t} g_{ij}=-2R_{ij}NEWLINE\]NEWLINE on a smooth, compact and \(n\)-dimensional Riemannian manifold \(M\). If the flow has uniformly bounded scalar curvature and develops type I singularities at \(T\), it is shown that suitable blow-ups of the evolving metrics converge in the pointed Cheeger-Gromov sense to a Gaussian shrinker by using Perelman's \(\mathcal W\)-functional. If the flow has uniformly bounded scalar curvature and develops type II singularities at \(T\), it is shown that suitable scalings of the potential functions in Perelman's entropy functional converge to a positive constant on a complete, Ricci flat manifold. As a consequence, 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 \(T\).
scientific article; zbMATH DE number 6029231