(K, N)-Convexity and the Curvature-Dimension Condition for Negative N

Ohta, Shin‐ichi

doi:10.1007/s12220-015-9619-1

Cited by 75 publications

(103 citation statements)

References 35 publications

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“…Remark. We can replace the CD p (K, ∞)-condition by the CD * p (K, N )-condition with N < 0 as defined by Ohta in [Oht16]. Indeed, following the proof of [Oht16,Theorem 4.8] gives a stronger variant of the Brunn-Minkowski inequality (replace A t by Γ t ) which for K = 0 and r = − 1 N > 0 says…”

Section: Non-degenericity Of γ Follows By Observing That For All Borementioning

confidence: 99%

Transport maps, non-branching sets of geodesics and measure rigidity

Kell

2017

Advances in Mathematics

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In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala-Sturm (Calc.Var.PDE 2014). In particular, it is shown that the qualitative non-degenericity condition introduced by Cavalletti-Huesmann (Ann. Inst. H. Poincar é Anal. Non Linéaire 2015) implies that any essentially non-branching metric measure space has a unique transport maps whenever initial measure is absolutely continuous. This generalizes a recently obtained result by Cavalletti-Mondino (Commun. Contemp. Math. 2017) on essentially non-branching spaces with the measure contraction condition MCP(K, N ).In the end we prove a measure rigidity result showing that any two essentially non-branching, qualitatively non-degenerate measures on a fixed metric spaces must be mutually absolutely continuous. This result was obtained under stronger conditions by Cavalletti-Mondino (Adv.Math. 2016). It applies, in particular, to metric measure spaces with generalized finite dimensional Ricci curvature bounded from below.

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Section: Non-degenericity Of γ Follows By Observing That For All Borementioning

confidence: 99%

Transport maps, non-branching sets of geodesics and measure rigidity

Kell

2017

Advances in Mathematics

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“…However, as soon as an α bn for some 0 < a < b < 3, observe that Finally, we apply the Lichnerowicz spectral-gap estimate from our previous work with A. Kolesnikov [14] (proved for weighted manifolds with convex boundaries, and also independently obtained by S.-I. Ohta [27] in the case of closed manifolds); the improvement in the range α ∈ (−1, 1) below is obtained by invoking the Maz'ya-Cheeger inequality as in [23,Theorem 6.1]. Let λ n,α x > 0 denote the spectral-gap of (S n , g, µ n,α x ), i.e.…”

Section: Y) < ε} Denotes the ε Extension Ofmentioning

confidence: 87%

Harmonic Measures on the Sphere via Curvature-Dimension

Milman¹

2017

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Since M is 1-dimensional and Ric N ≥ K, ψ satisfies the (K, N − 1)convexity condition in [Oh3]: ψ ′′ − (ψ ′ ) 2 N −1 ≥ K in the weak sense. Hence by [Oh3,(2.5…”

Section: Rigidity For Isoperimetric Inequality Of Negative Effective mentioning

confidence: 99%

Rigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds

Mai¹

2018

Geom Dedicata

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We study, on a weighted Riemannian manifold of Ric N ≥ K > 0 for N < −1, when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product R × cosh( √ K/(1−N )t) Σ n−1 of hyperbolic nature, where Σ n−1 is an (n − 1)-dimensional manifold with lower weighted Ricci curvature bound and R is equipped with a hyperbolic cosine measure. This is a similar phenomenon to the equality condition of Poincaré inequality. Moreover, every isoperimetric minimizer set is isometric to a half-space in an appropriate sense.In a weighted manifold, the Ricci curvature is modified into the weighted Ricci curvature Ric N involving a parameter N which is sometimes called the effective dimension. Recently, the developments on the curvature-dimension condition in the sense of Lott, Sturm and Villani have shed new light on the theory of curvature bounds for weighted manifolds. The curvature-dimension condition CD(K, N) is a synthetic notion of lower Ricci curvature bounds for metric measure spaces. The parameters K and N are usually regarded as "a lower bound of the Ricci curvature" and "an upper bound of the dimension", respectively. The roles of K and N are better understood when we consider a weighted Riemannian manifold (M, g, m):Geometric analysis for the weighted Ricci curvature Ric N (also called the Bakry-Émery-Ricci curvature) including the isoperimetric inequality has been intensively studied by Bakry and his collaborators in the framework of the Γ-calculus (see [BGL] and [BL]). Recently it turned out that there is a rich theory also for N ∈ (−∞, 1], though this range seems strange due to the above interpretation of N as an upper dimension bound. For examples, various Poincaré-type inequalities ([KM]), the curvature-dimension condition ([Oh3, Oh2]), the splitting theorem ([Wy]) were studied for N < 0 or N ≤ 1. In our previous paper [Mai], we studied the rigidity of the Poincaré inequality (spectral gap) under the condition Ric N ≥ K > 0 with N < −1, and showed that the sharp spectral gap is achieved only if the space is isometric to the warped product R × cosh( √ K/(1−N )t) Σ n−1 of hyperbolic nature. In this paper, we continue this study to the rigidity problem of the isoperimetric inequality.The isoperimetric inequality on weighted manifolds satisfying Ric N ≥ K and diam(M) ≤ D was studied in [Mi1] and [Mi2]. The isoperimetric inequality could also be verified in a gentle way called the needle decomposition on Riemannian manifolds developed by Klartag in [Kl]. The idea is to reduce a high dimensional inequality into its one dimensional version on geodesics, which is much easier to verify. Cavalletti and Mondino generalized this method to metric measure spaces satisfying CD(K, N) condition for N ∈ (1, ∞), and also established the rigidity result for the isoprimetic inequality in [CM]. In this paper, we use the needle decomposition method to consider the rigidity of the isoperimetric inequality under the condition Ric N ≥ K > 0 and N < −1. The splitting...

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(K, N)-Convexity and the Curvature-Dimension Condition for Negative N

Cited by 75 publications

References 35 publications

Transport maps, non-branching sets of geodesics and measure rigidity

Transport maps, non-branching sets of geodesics and measure rigidity

Harmonic Measures on the Sphere via Curvature-Dimension

Rigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds

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