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