On the conditions to extend Ricci flow(II)

TL;DR

This paper develops estimates under Ricci flow to analyze curvature blowup rates at singularities, establishing gap theorems that constrain the blowup behavior of Ricci and Riemann curvatures.

Contribution

It introduces new estimates under Ricci flow that lead to gap theorems for curvature blowup rates and shrinking Ricci solitons.

Findings

Curvature blowup rates are at least of type-I.

Gap theorems for Ricci and Riemann curvatures.

Constraints on singularity formation in Ricci flow.

Abstract

We develop some estimates under the Ricci flow and use these estimates to study the blowup rates of curvatures at singularities. As applications, we obtain some gap theorems: $\displaystyle \sup_X |Ric|$ and $\displaystyle \sqrt{\sup_X |Rm|} \cdot \sqrt{\sup_X |R|}$ must blowup at least at the rate of type-I. Our estimates also imply some gap theorems for shrinking Ricci solitons.