Sub-Riemannian interpolation inequalities

Barilari, Davide; Rizzi, Luca

doi:10.1007/s00222-018-0840-y

Cited by 35 publications

(51 citation statements)

References 55 publications

(74 reference statements)

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“…In particular, it is twice differentiable almost everywhere. Also, from Corollary 32 in [9], x = x 0 is in Cut 0 (x 0 ) if and only if r 0 fails to be semi-convex at x. Therefore, Cut 0 (x 0 ) has µ measure 0.…”

Section: Horizontal and Vertical Hessian And Laplacian Comparison Thementioning

confidence: 95%

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

et al. 2019

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We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical Laplacian comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical Laplacian comparison theorems for the Riemannian distances approximations. As a corollary we prove that, under suitable curvature conditions, sub-Riemannian Sasakian spaces are actually limits of Riemannian spaces satisfying a uniform measure contraction property.

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Section: Horizontal and Vertical Hessian And Laplacian Comparison Thementioning

confidence: 95%

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

et al. 2019

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“…Recently, interpolation inequalitiesà la Cordero-Erausquin-McCann-Schmuckenshläger [25] have been obtained, under suitable modifications, by Barilari and Rizzi [10] in the ideal sub-Riemannian setting and by Balogh, Kristly and Sipos [8] for the Heisenberg group. As a consequence, an increasing number of examples of spaces verifying MCP and not CD is at disposal, e.g.…”

Section: Isoperimetric Inequality Under Measure-contraction Propertymentioning

confidence: 99%

“…As a consequence, an increasing number of examples of spaces verifying MCP and not CD is at disposal, e.g. the Heisenberg group, generalized H-type groups, the Grushin plane and Sasakian structures (for more details, see [10]). In all the previous examples a sharp isoperimetric inequality is not at disposal yet; due to lack of regularity of minimizers, sharp isoperimetric inequality has been proved for subclasses of competitors having extra regularity or additional symmetries; in particular, Pansu Conjecture [43] is still unsolved.…”

Section: Isoperimetric Inequality Under Measure-contraction Propertymentioning

confidence: 99%

Isoperimetric inequality under Measure-Contraction property

Cavalletti

Santarcangelo

2019

Journal of Functional Analysis

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We prove that if (X, d, m) is an essentially non-branching metric measure space with m(X) = 1, having Ricci curvature bounded from below by K and dimension bounded above by N ∈ (1, ∞), understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequalityà la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained. model isoperimetric profile I CD K,N,D such that if a Riemannian manifold with density verifying CD(K, N ) has diameter at most D > 0, then the isoperimetric profile function of the weighted manifold is bounded from below by I CD K,N,D . After the works of Cordero-Erausquin-McCann-Schmuckenshläger [25], Otto-Villani [42] and von Renesse-Sturm [47], it was realized that the CD(K, ∞) condition in the smooth setting may be equivalently formulated synthetically as a certain convexity property of an entropy functional along W 2 Wasserstein geodesics (associated to L 2 -Optimal-Transport). This idea led Lott-Villani [36] and Sturm [51, 52], to propose a successful (and compatible with the classical one) synthetic definition of CD(K, N ) for a general (complete, separable) metric space (X, d) endowed with a (locally-finite Borel) reference measure m ("metricmeasure space", or m.m.s.); the theory of m.m.s.'s verifying CD(K, N ) has then extensively developed leading to a rich and fruitful approach to the geometry of m.m.s.'s by means of 3,4,27,26,5,38,29,33,13]. See also [1] for a recent account on the topic. Building on the work by Klartag [34] and the localization paradigm developed by Payne-Weinberger [44], Gromov-Milman [31] and Kannan-Lovász-Simonovits [32], the first author with Mondino [19] managed to extend Lévy-Gromov-Milman isoperimetric inequality to the class of essentially non-branching (see Section 2 for the definition) m.m.s.'s verifying CD(K, N ) with m(X) = 1; in particular [19] proves that

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“…Since complete

MCP (K, N)

spaces with

K > 0

are bounded, the Grushin plane or half‐planes can only satisfy

MCP (K, N)

for

K ⩽ 0

. The following fact was proved in [, Theorem 10]. Theorem The Grushin plane

double-struckG

, equipped with the Lebesgue measure, satisfies the

MCP (K, N)

if and only if

K ⩽ 0

and

N ⩾ 5

.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

A counterexample to gluing theorems for MCP metric measure spaces

Rizzi

2018

Bull. London Math. Soc.

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Perelman's doubling theorem asserts that the metric space obtained by gluing along their boundaries two copies of an Alexandrov space with curvature κ is an Alexandrov space with the same dimension and satisfying the same curvature lower bound. We show that this result cannot be extended to metric measure spaces satisfying synthetic Ricci curvature bounds in the MCP sense. The counterexample is given by the Grushin half-plane, which satisfies the MCP(0, N) if and only if N 4, while its double satisfies the MCP(0, N) if and only if N 5.

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Sub-Riemannian interpolation inequalities

Cited by 35 publications

References 55 publications

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

Isoperimetric inequality under Measure-Contraction property

A counterexample to gluing theorems for MCP metric measure spaces

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