• Corpus ID: 204575834

# Quantitative Obata's Theorem

@article{Cavalletti2019QuantitativeOT,
title={Quantitative Obata's Theorem},
author={Fabio Cavalletti and Andrea Mondino and Daniele Semola},
journal={arXiv: Functional Analysis},
year={2019}
}
• Published 15 October 2019
• Mathematics
• arXiv: Functional Analysis
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…
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## References

SHOWING 1-10 OF 61 REFERENCES

Abstract We prove Obata’s rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally,we show that a lower bound K for the generalizedHessian of a
• Mathematics
Communications on Pure and Applied Mathematics
• 2018
On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the
This article concerns several geometric properties of metric measure spaces satisfying the measure contraction property (MCP), which can be considered as a generalized notion of lower Ricci curvature
• Mathematics
Rendiconti Lincei - Matematica e Applicazioni
• 2018
We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of
• Mathematics
• 2010
A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved
• Mathematics
• 2013
We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Émery (via energy and $$\Gamma _2$$Γ2-calculus) in complete
• Mathematics
• 2015
The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a sharp quantitative
We prove that in metric measure spaces where the entropy functional is K -convex along every Wasserstein geodesic any optimal transport between two absolutely continuous
• Mathematics
• 2004
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the
We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and