Ricci flow on open 3–manifolds and positive scalar curvature

Bessières, Laurent¹ ; Besson, Gérard¹ ; Maillot, Sylvain²

¹ Institut Fourier, UMR CNRS 5582 Université de Grenoble I, BP 74, 100 rue des maths, 38402 Saint Martin d’Hères, France
² Institut de Mathématiques et de Modélisation de Montpellier (I3M), UMR CNRS 5149 Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34095 Montpellier, France

Geometry & topology, Tome 15 (2011) no. 2, pp. 927-975.

Résumé

We show that an orientable $3$–dimensional manifold $M$ admits a complete riemannian metric of bounded geometry and uniformly positive scalar curvature if and only if there exists a finite collection $\mathcal{ℱ}$ of spherical space-forms such that $M$ is a (possibly infinite) connected sum where each summand is diffeomorphic to $S^2\times S^1$ or to some member of $\mathcal{ℱ}$. This result generalises G Perelman’s classification theorem for compact $3$–manifolds of positive scalar curvature. The main tool is a variant of Perelman’s surgery construction for Ricci flow.

Détail

DOI : 10.2140/gt.2011.15.927

Keywords: Ricci flow, three-dimensional topology

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