Ricci flow with surgery on manifolds with positive isotropic curvature

Abstract

We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short).

In the first part of this paper, we prove new curvature pinching estimates that ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are weakly ${\rm PIC}2$. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$.

In the second part, we develop a theory of ancient solutions that have bounded curvature, are $\kappa $-noncollapsed, are weakly ${\rm PIC}2$, and are uniformly PIC. This is an extension of Perelman’s work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly PIC1.

In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension $n \geq 12$. This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman’s surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension $n \geq 12$ that do not contain nontrivial incompressible $(n-1)$-dimensional space forms.