Sub-Riemannian interpolation inequalities
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 [Calc. Var. PDE (2018) 57: 61], 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.
Keywords
Cite
@article{arxiv.1705.05380,
title = {Sub-Riemannian interpolation inequalities},
author = {Davide Barilari and Luca Rizzi},
journal= {arXiv preprint arXiv:1705.05380},
year = {2018}
}
Comments
43 pages. v2: typo corrected in the statement of Theorem 10. v3: updated references, added Corollary 11, typos corrected. v4: major revision, improved exposition and updated references, added section 7.4 on Sasakian examples. v5, v6 minor corrections. To appear on Inventiones Mathematicae