Quantitative Obata's Theorem

Andrea Mondino; Daniele Semola; Fabio Cavalletti

Quantitative Obata's Theorem

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arxiv 1910.06637 v3 pith:HGN7HFHA submitted 2019-10-15 math.FA math.DGmath.MG

Fabio Cavalletti , Andrea Mondino , Daniele Semola This is my paper · ORCID

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keywords quantitativefunctionsobataobtainedtheoremanalysisboundedcompared

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

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Lipschitz continuity of harmonic maps between ${\rm RCD}(K,N)$ spaces and ${\rm CAT}(\kappa)$ spaces
math.AP 2026-07 accept novelty 5.0 of 10

Energy-minimizing harmonic maps from RCD(K,N) domains into small balls in CAT(κ) spaces are locally Lipschitz, completing the singular Bochner–Eells–Sampson picture.

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