Topology of $3$-manifolds with uniformly positive scalar curvature

TL;DR

This paper classifies non-compact 3-manifolds with complete metrics of uniformly positive scalar curvature, showing they are homeomorphic to sums of spherical 3-manifolds and products of circles and spheres.

Contribution

It provides a complete topological classification of 3-manifolds admitting such metrics, including those with boundary, extending previous results in scalar curvature geometry.

Findings

Characterization of 3-manifolds with positive scalar curvature

Extension to manifolds with boundary and mean convex boundary

Description of manifold decomposition into known topological pieces

Abstract

In this article, we classify (non-compact) $3$-manifolds with uniformly positive scalar curvature. Precisely, we show that an oriented $3$-manifold has a complete metric with uniformly positive scalar curvature if and only if it is homeomorphic to an (possibly) infinite connected sum of spherical $3$-manifolds and some copies of $\mathbb{S}^1\times \mathbb{S}^2$. Further, we study an oriented $3$-manifold with mean convex boundary and with uniformly positive scalar curvature. If the boundary is a disjoint union of closed surfaces, then the manifold is an (possibly) infinite conned sum of spherical $3$-manifolds, some handlebodies and some copies of $\mathbb{S}^1\times \mathbb{S}^2$.