September 2019 Ricci flow with surgery on manifolds with positive isotropic curvature
Simon Brendle
Author Affiliations +
Ann. of Math. (2) 190(2): 465-559 (September 2019). DOI: 10.4007/annals.2019.190.2.2


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 $n\ge 12$, we show that blow-up limits are weakly PIC2. 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 PIC2, 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 $n\ge 12$. 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 $n \ge 12$ that do not contain nontrivial incompressible $(n-1)$-dimensional space forms.


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Simon Brendle. "Ricci flow with surgery on manifolds with positive isotropic curvature." Ann. of Math. (2) 190 (2) 465 - 559, September 2019.


Published: September 2019
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2019.190.2.2

Primary: 53C44

Keywords: curvature pinching , Ricci flow , surgery

Rights: Copyright © 2019 Department of Mathematics, Princeton University


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Vol.190 • No. 2 • September 2019
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