A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces

Kitabeppu, Yu

doi:10.1007/s11118-018-9708-4

Cited by 26 publications

(22 citation statements)

References 37 publications

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“…To let the picture about the different notions of dimension introduced in the literature so far be more complete, we also point out that n is also the dimension of (X, d, m) according to [K18, Definition 4.1]. Indeed, as it is observed in [K18,Remark 4.14], if R n is the unique regular set of positive measure, the results of [K18] grant that it is also the non empty regular set of maximal dimension.…”

Section: Hence Settingmentioning

confidence: 99%

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Bruè

Semola

2019

Comm Pure Appl Math

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We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N ) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K, N ) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting. * Scuola Normale Superiore, elia.brue@sns.it. † Scuola Normale Superiore, daniele.semola@sns.it.

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Section: Hence Settingmentioning

confidence: 99%

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Bruè

Semola

2019

Comm Pure Appl Math

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“…In [29] Kitabeppu defined the Analytic dimension, dimX , of an RCD * (K, N ) space as the largest k ∈ [1, N ] ∩ N such that m(R k ) > 0. For both Riemannian manifolds and Alexandrov spaces it is easy to see that the Analytic dimension coincides with the more classical notions used in the literature.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Kitabeppu introduced in [29] a new concept of dimension that coincides with previous definitions made by Colding and Naber for Ricci limit spaces [12] and by Han [25]. [29].) Let (X, d, m) be an RCD * (K, N ) space.…”

Section: 5mentioning

confidence: 98%

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Invariant Measures and Lower Ricci Curvature Bounds

Santos-Rodríguez

2019

Potential Anal

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Given a metric measure space (X, d, m) that satisfies the Riemannian Curvature Dimension condition, RCD * (K, N ), and a compact subgroup of isometries G ≤ Iso(X) we prove that there exists a G−invariant measure, mG, equivalent to m such that (X, d, mG) is still a RCD * (K, N ) space. We also obtain some applications to Lie group actions on RCD * (K, N ) spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries.

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“…The next theorem follows from a combination of [Kit17] and [DePhG18]. For the reader's convenience we give a proof: To conclude this section, we introduce a compactness result for non-collapsed RCD spaces with respect to pmGH convergence, which is proved in [DePhG18, Thm.1.2, Thm.1.3] (Compare with Theorem 1.1); Theorem 1.11 (Compactness of non-collapsed RCD spaces).…”

Section: Preliminarymentioning

confidence: 99%

Sphere theorems for RCD and stratified spaces

Honda¹,

Mondello²

2021

Annali Scuola Normale Superiore - Classe Di Scienze

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We prove topological sphere theorems for RCD(n − 1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.the singular space associated to a static triple [A17]; in Kähler geometry, Kähler-Einstein manifolds with edge singularities of cone angle smaller than 2π along a smooth divisor [JMR16].We are now in a position to state the main result of the paper:Theorem A (Topological sphere theorem for RCD spaces, I). For all n ∈ N ≥2 there exists a positive constant ǫ n > 0 such that if a compact metric space (X, d) satisfies that rad(X, d) ≥ π − ǫ n and that (X, d, m) is a RCD(n − 1, n) space for some Borel measure m on X with full support, then X is homeomorphic to the n-dimensional sphere.This seems the first topological sphere theorem in the RCD theory. We emphasize again that the theorem states that although there is a flexibility on the choice of m, the topological structure is uniquely determined. Note that in the previous theorem one cannot replace the radius by the diameter of the space. Indeed, for any ε > 0 Anderson constructed in [A90] manifolds of even dimension n ≥ 4, with Ricci tensor bounded below by n − 1 and diameter larger than π − ǫ, which are not homeomorphic to the sphere. Similar examples can be found in [O91] by Otsu.In order to introduce an application, let us recall a result of Petersen [Pet99]; for a closed n-dimensional Riemannian manifold (M n , g) with Ric g M n ≥ n − 1 the following two conditions are equivalent quantitatively:(1) The (n + 1)-th eigenvalue of the Laplacian is close to n (2) The radius is close to π.In particular if the one of them above holds, then M n is diffeomorphic to S n .Note that even in the RCD-setting, the above equivalence is justified by the spectral convergence result of Gigli-Mondino-Savaré [GMS13] and the rigidity results of Ketterer [K15b]. In particular we have the following;

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A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces

Cited by 26 publications

References 37 publications

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Invariant Measures and Lower Ricci Curvature Bounds

Sphere theorems for RCD and stratified spaces

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