Ricci flow on open 3–manifolds and positive scalar curvature

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 F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S-2 x S-1 or to some member of 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.