Torus stability under Kato bounds on the Ricci curvature

2022/07/12 by Carron, Gilles, Mondello, Ilaria, Tewodrose, David
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.05419

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.

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