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 $\mu$-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.

14 pages, comments welcome. v2: Added a remark to clarify basepoint dependency in Definition 1.1 and fixed typos. arXiv admin note: text overlap with arXiv:2407.07198 by other authors