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

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 Sardà, and Sabourau, and partially answers a conjecture of Gromov. The main tool is a new exhaustion result using $μ$-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.