Optimal decay constant for complete manifolds of positive scalar curvature with quadratic decay

TL;DR

This paper establishes the optimal decay constant for positive scalar curvature metrics on complete 3-manifolds, showing they decompose into spherical and product summands, and extends results to higher dimensions with topological obstructions.

Contribution

It proves the sharp decay constant threshold for positive scalar curvature on 3-manifolds and introduces a new exhaustion technique using μ-bubbles, extending to higher dimensions.

Findings

Decay constant threshold of 2/3 is sharp for 3-manifolds.

Manifolds with decay constant > 2/3 decompose into spherical and S^2×S^1 summands.

Topological obstructions are identified in dimensions 4 and 5 for certain decay conditions.

Abstract

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has at most $C$-quadratic decay at infinity for some $C > \frac{2}{3}$, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and $\mathbb{S}^2\times \mathbb{S}^1$ summands. Consequently, $M$ carries a complete Riemannian metric of uniformly positive scalar curvature. The decay constant $\frac{2}{3}$ is sharp, as demonstrated by metrics on $\mathbb{R}^2 \times \mathbb{S}^1$. This improves a result of Balacheff, Gil Moreno de Mora Sard\`a, and Sabourau, and partially answers a conjecture of Gromov. The main tool is a new exhaustion result using $\mu$-bubbles. In dimensions $n = 4, 5$, we further extend results of Chodosh--Maximo--Mukherjee and Sweeney, and obtain topological obstructions to the existence of a complete Riemannian metric whose scalar curvature is positive and has at most $C$-quadratic decay at infinity for some $C > \frac{n-1}{n}$ on certain noncompact contractible $n$-manifolds.