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Ricci flow with surgery on four-manifolds with positive isotropic curvature. (English) Zbl 1103.53036
The authors study the Ricci flow on a compact \(4\)-manifold with positive isotropic curvature and with no essential incompressible space form, establishing a long-time existence result of the Ricci flow with surgery on \(4\)-manifolds. As a consequence, they obtain a complete proof of the main theorem of R. S. Hamilton in [Commun. Anal. Geom. 5, No. 1, 1–92 (1997; Zbl 0892.53018)]. In the course of the proof, they also provide, up to slight modifications, the details of Section 1 to Section 5 of G. Perelman’s second paper [arXiv e-print service, Cornell University Library, Paper No. 0307245, 7 p., electronic only (2003)] on the Ricci flow to approach the PoincarĂ© conjecture.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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