Calculus of Variations and Geometric Measure Theory
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D. Barilari - L. Rizzi

Sub-Riemannian interpolation inequalities: ideal structures

created by rizzi1 on 18 May 2017

[BibTeX]

Preprint

Inserted: 18 may 2017
Last Updated: 18 may 2017

Year: 2017

ArXiv: 1705.05380 PDF

Abstract:

We prove that ideal sub-Riemannian manifolds (i.e., admitting no non-trivial abnormal minimizers) support interpolation inequalities for optimal transport. A key role is played by sub-Riemannian Jacobi fields and distortion coefficients, whose properties are remarkably different with respect to the Riemannian case. As a byproduct, we characterize the cut locus as the set of points where the squared sub-Riemannian distance fails to be semiconvex, answering to a question raised by Figalli and Rifford in Geom. Funct. Anal. (2010) 20: 124. As an application, we deduce sharp and intrinsic Borell-Brascamp-Lieb and geodesic Brunn-Minkowski inequalities in the aforementioned setting. For the case of the Heisenberg group, we recover in an intrinsic way the results recently obtained by Balogh, Krist\'aly and Sipos in arXiv:1605.06839, and we extend them to the class of generalized H-type Carnot groups. Our results do not require the distribution to have constant rank, yielding for the particular case of the Grushin plane a sharp measure contraction property and a sharp Brunn-Minkowski inequality.


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