Calculus of Variations and Geometric Measure Theory

G. Carron - I. Mondello - D. Tewodrose

Stability of the Betti number under Kato bounds on the Ricci curvature

created by tewodrose on 12 Jul 2022
modified on 12 Mar 2023


Published Paper

Inserted: 12 jul 2022
Last Updated: 12 mar 2023

Journal: Journal of the London Mathematical Society
Volume: 107(3)
Pages: 943--97
Year: 2023

ArXiv: 2207.05419v2 PDF


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.