Geometry & Topology

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

Laurent Bessières, Gérard Besson, and Sylvain Maillot

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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 of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S2×S1 or to some member of . 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.

Article information

Geom. Topol., Volume 15, Number 2 (2011), 927-975.

Received: 10 February 2010
Revised: 25 March 2011
Accepted: 8 May 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 57M50: Geometric structures on low-dimensional manifolds

Ricci flow three-dimensional topology


Bessières, Laurent; Besson, Gérard; Maillot, Sylvain. Ricci flow on open $3$–manifolds and positive scalar curvature. Geom. Topol. 15 (2011), no. 2, 927--975. doi:10.2140/gt.2011.15.927.

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