Convergence of Ricci-limit spaces under bounded Ricci curvature and local covering geometry I

TL;DR

This paper extends convergence theorems for manifolds with bounded Ricci curvature, establishing optimal regularity and generalizing Fukaya's fibration theorem to Ricci-limit spaces with improved regularity.

Contribution

It introduces new $C^{1,eta}$ regularity results for Ricci-limit spaces and generalizes Fukaya's fibration theorem to these spaces, enhancing previous results.

Findings

Established $C^{1,eta}$ regularity for Ricci-limit spaces.

Generalized Fukaya's fibration theorem to $C^{1,eta}$ limit spaces.

Improved understanding of collapsed manifolds with bounded Ricci curvature.

Abstract

We extend Cheeger-Gromov's and Anderson's convergence theorems to regular limit spaces of manifolds with bounded Ricci curvature and local covering geometry, by establishing the $C^{1,\alpha}$-regularities that are the best one may expect on those Ricci-limit spaces. As an application we prove an optimal generalization of Fukaya's fibration theorem on collapsed manifolds with bounded Ricci curvature, which also improves the original version to $C^{1,\alpha}$ limit spaces.