Limits of manifolds with a Kato bound on the Ricci curvature. II

TL;DR

This paper proves that limits of Riemannian manifolds with Ricci curvature bounds satisfying a Kato condition are rectifiable, and under stronger assumptions, their regular parts are locally $ ext{C}^eta$-manifolds.

Contribution

It establishes rectifiability and regularity of limit spaces under Kato bounds on Ricci curvature, extending geometric analysis in metric measure spaces.

Findings

Limit spaces are rectifiable.

Regular parts are locally $ ext{C}^eta$-manifolds under strong bounds.

Results extend understanding of Ricci curvature limits.

Abstract

We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally show that for any $\alpha \in (0,1)$ the regular part of the space lies in an open set with the structure of a $\mathcal{C}^\alpha$-manifold.