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Quantitative Obata's theorem

Fabio Cavalletti, Andrea Mondino and Daniele Semola

Vol. 16 (2023), No. 6, 1389–1431

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.

quantitative inequalities, Obata's theorem, Ricci curvature, spectral gap
Mathematical Subject Classification
Primary: 53C23, 58J50
Received: 24 July 2021
Revised: 20 December 2021
Accepted: 31 January 2022
Published: 23 August 2023
Fabio Cavalletti
Scuola Internazionale Superiore di Studi Avanzati
Andrea Mondino
Mathematical Institute
University of Oxford
United Kingdom
Daniele Semola
Scuola Normale Superiore

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