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, Kristály 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.

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

References in corpus (9)

• Curvature: a variational approach
• On the Hausdorff volume in sub-Riemannian geometry
• On the cut locus of free, step two Carnot groups
• Sharp measure contraction property for generalized H-type Carnot groups
• On Jacobi fields and canonical connection in sub-Riemannian geometry
• Comparison theorems for conjugate points in sub-Riemannian geometry
• Measure contraction properties of Carnot groups
• On the subRiemannian cut locus in a model of free two-step Carnot group
• A counterexample to gluing theorems for MCP metric measure spaces

Cited by in corpus (22)

• Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries
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• Bakry-Émery curvature and model spaces in sub-Riemannian geometry
• Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry
• Distortion coefficients of the $\alpha$-Grushin plane
• Sharp Poincaré inequalities under Measure Contraction Property
• Affine connections and curvature in sub-Riemannian geometry
• Comparison theorems on H-type sub-Riemannian manifolds
• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Horizontal semiconcavity for the square of Carnot-Carathéodory distance on step 2 Carnot groups and applications to Hamilton-Jacobi equations
• Curvature exponent and geodesic dimension on Sard-regular Carnot groups
• Uniform, localized asymptotics for sub-Riemannian heat kernels, their logarithmic derivatives, and associated diffusion bridges
• Sub-Riemannian geometry on some step-two Carnot groups
• On gradient estimates of the heat semigroups on step-two Carnot groups
• Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures
• Isoperimetric inequality in noncompact MCP spaces
• On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: asymptotic volume ratio, volume entropy and rigidity
• SubRiemanniann structures do not satisify Riemannian Brunn--Minkowski inequalities
• Mahler type theorem and non-collapsed limits of sub-Riemannian compact Heisenberg manifolds
• Geodesic fields for Pontryagin type $C^0$-Finsler manifolds
• The sharp $p$-Poincaré inequality under the measure contraction property
• Isometric rigidity of the Wasserstein space ${\mathcal{W}}_1(\mathbf{G})$ over Carnot groups