Geometric Analysis Seminar: Shuli Chen (UChicago)

Title: 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 > ²⁄₃, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and 𝕊² × 𝕊¹ summands. Consequently, M carries a complete Riemannian metric of uniformly positive scalar curvature. The decay constant ²⁄₃ is sharp, as demonstrated by metrics on ℝ² × 𝕊¹. 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.