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 > 23 C > 2 3, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and S² S¹ S 2 × S 1 summands. Consequently, M carries a complete Riemannian metric of uniformly positive scalar curvature. The decay constant 23 2 3 is sharp, as demonstrated by metrics on R² S¹ R 2 × 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 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 > n-1n C > n - 1 n on certain noncompact contractible n -manifolds.
Shuli Chen (Tue,) studied this question.