Calculus of Variations and Geometric Measure Theory

F. Cavalletti - A. Mondino - D. Semola

Quantitative Obata's Theorem

created by cavallett on 16 Oct 2019
modified by semola on 28 Aug 2023


Published Paper

Inserted: 16 oct 2019
Last Updated: 28 aug 2023

Journal: Analysis & PDE
Volume: 16
Number: 6
Pages: 1389–1431
Year: 2023
Doi: 10.2140/apde.2023.16.1389


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 non-smooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport-rays decompositions obtained by the localization method.