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

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.