A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators

Dario Prandi (1) , Luca Rizzi (2) , Marcello Seri (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 Department of Mathematics, University of Reading

We prove a sub-Riemannian version of the classical Santal\'o formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. Our construction works under quite general conditions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g. CR and quaternionic contact manifolds with symmetries) and any Carnot group. A key ingredient is a "reduction procedure" that allows to consider only a simple subset of sub-Riemannian geodesics. As an application, we derive (p-)Hardy-type and isoperimetric-type inequalities for a compact domain M with piecewise C^2 boundary. Moreover, we prove a universal (i.e. curvature independent) lower bound for the first Dirichlet eigenvalue λ1(M) of the intrinsic sub-Laplacian, λ1(M)≥kπ2L2, in terms of the rank k of the distribution and the length L of the longest reduced sub-Riemannian geodesic contained in M. All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations: S1↪S2d+1→pCPd,S3↪S4d+3→pHPd,d≥1, where the sub-Laplacian is the standard hypoelliptic operator of CR and quaternionic contact geometries, L=π and k=2d or 4d, respectively.

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

Dario Prandi, Luca Rizzi, Marcello Seri. A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators. 2015. ⟨hal-01221668v2⟩