Positive Scalar Curvature Meets Ricci Limit Spaces

TL;DR

This paper explores how positive scalar curvature constrains the geometric structure of Ricci limit spaces, establishing bounds on splitting, diameter, and volume growth in non-collapsed limits.

Contribution

It proves that positive scalar curvature limits the splitting of Ricci limit spaces and provides bounds on diameter and volume growth, advancing understanding of geometric constraints.

Findings

Limit spaces split at most n-2 lines or R-factors.

Uniform upper bound on diameter of non-splitting factors.

Volume gap and growth estimates for geodesic balls.

Abstract

We investigate the influence of uniformly positive scalar curvature on the size of a non-collapsed Ricci limit space coming from a sequence of $n$-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature. We prove that such a limit space splits at most $n-2$ lines or $\mathbb{R}$-factors. When this maximal splitting occurs, we obtain a uniform upper bound on the diameter of the non-splitting factor. Moreover, we obtain a volume gap estimate and a volume growth order estimate of geodesic balls on such manifolds.