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

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 > 2/3 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 2/3 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 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-1/n C > n - 1 n on certain noncompact contractible n -manifolds.