Positive Scalar Curvature Meets Ricci Limit Spaces

Abstract

We investigate how the positive scalar curvature controls the size of a Ricci limit space when it comes from a sequence of $n$-manifolds with non-negative Ricci curvature and strictly positive scalar curvature lower bound. We prove such a limit space can split off $\mathbb{R}^{n-2}$ at most, and when the maximal splitting happens, the other non-splitting factor has an explicit uniform diameter upper bound. Besides, we study some other consequences of having positive scalar curvature for manifolds using Ricci limit spaces techniques, for instance volume gap estimates and volume growth order estimates.Comment: 22 pages. Some conditions added to the theorem 1.1 due to a gap in the original proof, and the proof is slightly changed accordingly. A corollary about the first Betti number is adde