Optimal diameter estimates of three-dimensional Ricci limit spaces

TL;DR

This paper establishes optimal diameter bounds for three-dimensional Ricci limit spaces with non-negative Ricci curvature, demonstrating how positive scalar curvature influences their geometric structure and extending results to Alexandrov spaces.

Contribution

It proves that positive scalar curvature passes to Ricci limit spaces when splitting off a line, leading to optimal diameter bounds and extending results to Alexandrov spaces.

Findings

Positive scalar curvature passes to Ricci limit spaces with a line split.

Established an optimal Bonnet-Myers type diameter bound.

Extended diameter estimates to Alexandrov spaces of non-negative curvature.

Abstract

In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature.