Torus stability under Kato bounds on the Ricci curvature

TL;DR

This paper proves that closed Riemannian manifolds with small Kato-bounded Ricci curvature and maximal first Betti number are geometrically close to, or even diffeomorphic to, flat tori, extending classical stability results.

Contribution

It establishes new stability and diffeomorphism results for manifolds with Ricci curvature bounds in the Kato sense, generalizing previous lower Ricci bounds.

Findings

Manifolds are Gromov-Hausdorff close to flat tori.

Under stronger assumptions, manifolds are diffeomorphic to tori.

Extends classical Ricci curvature stability results.

Abstract

We show two stability results for a closed Riemannian manifold whose Ricci curvature is small in the Kato sense and whose first Betti number is equal to the dimension. The first one is a geometric stability result stating that such a manifold is Gromov-Hausdorff close to a flat torus. The second one states that, under a stronger assumption, such a manifold is diffeomorphic to a torus: this extends a result by Colding and Cheeger-Colding obtained in the context of a lower bound on the Ricci curvature.