Complete 3-manifolds of positive scalar curvature with quadratic decay

TL;DR

This paper proves that orientable 3-manifolds with complete metrics of positive scalar curvature and quadratic decay at infinity decompose into spherical and $ ext{S}^2 imes ext{S}^1$ summands, extending previous results and addressing a conjecture of Gromov.

Contribution

It introduces a topological approach to decompose such 3-manifolds under quadratic decay conditions, generalizing Gromov and Wang's theorem.

Findings

Manifolds decompose into spherical and $ ext{S}^2 imes ext{S}^1$ summands.

Quadratic decay rate of scalar curvature is shown to be optimal.

Decomposition holds under weaker conditions without scalar curvature assumptions.

Abstract

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has a subquadratic decay at infinity, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and $\mathbb{S}^2 \times \mathbb{S}^1$ summands. This generalises a theorem of Gromov and Wang by using a different, more topological, approach. As a result, the manifold $M$ carries a complete Riemannian metric of uniformly positive scalar curvature, which partially answers a conjecture of Gromov. More generally, the topological decomposition holds without any scalar curvature assumption under a weaker condition on the filling discs of closed curves in the universal cover based on the notion of fill radius. Moreover, the decay rate of the scalar curvature is optimal in this decomposition theorem. Indeed, the manifold $\mathbb{R}^2 \times \mathbb{S}^1$ supports a complete metric of positive scalar curvature with exactly quadratic decay, but does not admit a decomposition as a connected sum.