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.
