Sub-Riemannian interpolation inequalities

Davide Barilari (1) , Luca Rizzi (2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 IF - Institut Fourier

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 Rie-mannian 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 [FR10]. 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, Kristály and Sipos in [BKS16], 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.

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Optimisation et contrôle [math.OC]

Davide Barilari, Luca Rizzi. Sub-Riemannian interpolation inequalities. Inventiones Mathematicae, 2019, 215 (3), pp.977-1038. ⟨10.1007/s00222-018-0840-y⟩. ⟨hal-01524541⟩