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Quantitative Obata's
theorem
Fabio Cavalletti, Andrea Mondino and Daniele Semola |
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Vol. 16 (2023), No. 6, 1389–1431
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Abstract |
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We prove a quantitative version of Obata’s theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the general framework of (possibly nonsmooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport-ray decompositions obtained by the localization method. |
Keywords
quantitative inequalities, Obata's theorem, Ricci
curvature, spectral gap
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Mathematical Subject Classification
Primary: 53C23, 58J50
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Milestones
Received: 24 July 2021
Revised: 20 December 2021
Accepted: 31 January 2022
Published: 23 August 2023
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Authors |
| Fabio Cavalletti | |
| Scuola Internazionale Superiore di
Studi Avanzati Trieste Italy |
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| Andrea Mondino | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
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| Daniele Semola | |
| Scuola Normale Superiore Pisa Italy |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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