Torus stability under Kato bounds on the Ricci curvature

Gilles Carron, Ilaria Mondello, David Tewodrose

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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.

Original languageEnglish
Pages (from-to)943-972
Number of pages30
JournalJournal of the London Mathematical Society
Volume107
Issue number3
DOIs
Publication statusPublished - Mar 2023

Bibliographical note

Funding Information:
The authors thank the anonymous referee for constructive comments including Remark 3.3 . They are partially supported by the ANR grant ANR‐17‐CE40‐0034: CCEM. The first author is also partially supported by the ANR grant ANR‐18‐CE40‐0012: RAGE. The third author is supported by Laboratoire de Mathématiques Jean Leray via the project Centre Henri Lebesgue ANR‐11‐LABX‐0020‐01, and by Fédération de Recherche Mathématiques de Pays de Loire via the project Ambition Lebesgue Loire.

Funding Information:
The authors thank the anonymous referee for constructive comments including Remark 3.3. They are partially supported by the ANR grant ANR-17-CE40-0034: CCEM. The first author is also partially supported by the ANR grant ANR-18-CE40-0012: RAGE. The third author is supported by Laboratoire de Mathématiques Jean Leray via the project Centre Henri Lebesgue ANR-11-LABX-0020-01, and by Fédération de Recherche Mathématiques de Pays de Loire via the project Ambition Lebesgue Loire.

Publisher Copyright:
© 2022 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

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