• 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}
}
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… 
View PDF on arXiv
6 Citations
Highly Influential Citations
1
Background Citations
2
Methods Citations
2

6 Citations

Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp…

Stability estimates for the sharp spectral gap bound under a curvature-dimension condition

We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing…

Quantitative estimates for the Bakry–Ledoux isoperimetric inequality II

Concerning quantitative isoperimetry for a weighted Riemannian manifold satisfying Ric∞⩾1$\mathop {\mathrm{Ric}}\nolimits _{\infty } \geqslant 1$ , we give an L1$L^1$ ‐estimate exhibiting that the…

Stability of Eigenvalues and Observable Diameter in RCD ( 1 , ∞ ) Spaces

We study stability of the spectral gap and observable diameter for metric measure spaces satisfying the RCD ( 1 , ∞ ) condition. We show that if such a space has an almost maximal spectral gap, then…

Stability of Eigenvalues and Observable Diameter in RCD$$(1, \infty )$$ Spaces

We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1,∞) condition. We show that if such a space has an almost maximal spectral gap, then it…

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

We prove that if M is a closed n-dimensional Riemannian manifold, n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

References

SHOWING 1-10 OF 61 REFERENCES

Obata’s Rigidity Theorem for Metric Measure Spaces

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…

Quantitative Isoperimetry à la Levy‐Gromov

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…

Products, cones, and suspensions of spaces with the measure contraction property

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…

Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds

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…

A mass transportation approach to quantitative isoperimetric inequalities

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…

On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

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…

Faber–Krahn inequalities in sharp quantitative form

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…

Non-branching geodesics and optimal maps in strong C D ( K , ∞ ) -spaces

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…

Ricci curvature for metric-measure spaces via optimal transport

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…

Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition

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…
...