Ricci flow with surgery on four-manifolds with positive isotropic curvature. (English) Zbl 1103.53036

Chen, Bing-Long; Zhu, Xi-Ping

J. Differ. Geom. 74, No. 2, 177-264 (2006).

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 Poincaré conjecture.

Reviewer: Alberto Parmeggiani (Bologna)

MSC:

53C44

Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

Ricci flow; surgery; four-manifolds; isotropic curvature