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 $\mathcal{ℱ}$ 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.