Uniqueness of the Ricci flow on complete noncompact manifolds. (English) Zbl 1104.53032

Summary: We study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. We establish a long-time existence result of the Ricci flow with surgery on four-dimensional manifolds. As a consequence, we obtain a complete proof to the main theorem of R. S. Hamilton in [Commun. Anal. Geom. 5, No. 1, 1–92 (1997; Zbl 0892.53018)]. During the proof we have actually provided, up to slight modifications, all necessary details for the part form Section 1 to Section 5 of Perelman’s second paper on the Ricci flow to approach the Poincaré conjecture.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)


Zbl 0892.53018
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