Ricci flow on open 3-manifolds and positive scalar curvature

TL;DR

This paper characterizes open 3-manifolds with complete metrics of positive scalar curvature as connected sums of specific spherical space-forms and S2xS1, extending Perelman's compact case classification.

Contribution

It generalizes Perelman's classification theorem to open 3-manifolds using a modified Ricci flow surgery approach.

Findings

Characterization of open 3-manifolds with positive scalar curvature

Extension of Perelman's classification to non-compact cases

Development of a variant of Ricci flow surgery for open manifolds

Abstract

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive 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 S2xS1 or to some mem- ber 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.