#### Volume 15, issue 2 (2011)

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Ricci flow on open $3$–manifolds and positive scalar curvature

### Laurent Bessières, Gérard Besson and Sylvain Maillot

Geometry & Topology 15 (2011) 927–975
##### Abstract

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 $\mathsc{ℱ}$ of spherical space-forms such that $M$ is a (possibly infinite) connected sum where each summand is diffeomorphic to ${S}^{2}×{S}^{1}$ or to some member of $\mathsc{ℱ}$. 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.

##### Keywords
Ricci flow, three-dimensional topology
##### Mathematical Subject Classification 2000
Primary: 53C21, 53C44, 57M50
##### Publication
Revised: 25 March 2011
Accepted: 8 May 2011
Published: 18 June 2011
Proposed: David Gabai
Seconded: Peter Teichner, Gang Tian
##### Authors
 Laurent Bessières Institut Fourier UMR CNRS 5582 Université de Grenoble I BP 74 100 rue des maths 38402 Saint Martin d’Hères France http://www-fourier.ujf-grenoble.fr/~lbessier/ Gérard Besson Institut Fourier UMR CNRS 5582 Université de Grenoble I BP 74 100 rue des maths 38402 Saint Martin d’Hères France http://www-fourier.ujf-grenoble.fr/~besson/ Sylvain Maillot Institut de Mathématiques et de Modélisation de Montpellier (I3M) UMR CNRS 5149 Université Montpellier 2 Case Courrier 051 Place Eugène Bataillon 34095 Montpellier France http://www.math.univ-montp2.fr/~maillot/