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Geometric and spectral estimates based on spectral Ricci curvature assumptions

Gilles Carron and Christian Rose


We obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian ( M n , g ) manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator Δ + ( n - 2 ) ρ is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on the Ricci curvature. We also obtain the Kato condition for the Ricci curvature under purely geometric assumptions.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: CCEM-17-CE40-0034

Award Identifier / Grant number: ANR-11-LABX-0020-01

Funding statement: Gilles Carron was partially supported by the ANR grant CCEM-17-CE40-0034 and he wants to thank the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment.


Christian Rose wants to thank the regional project DéfiMaths and Gilles Carron for their hospitality during his stay in Nantes, where parts of this work had been done. We also want to thank the referee for carefully reading the manuscript and for providing suggestions that improved the presentation.


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Received: 2018-10-10
Revised: 2020-05-30
Published Online: 2020-08-05
Published in Print: 2021-03-01

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