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


We present a sub-Riemannian version of the classical Santaló formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. This 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 Lipschitz boundary and negligible characteristic set. Moreover, we prove a universal (i.e. curvature independent) lower bound for the first Dirichlet eigenvalue $\lambda_1(M)$ of the intrinsic sub-Laplacian,$$\lambda_1(M) \geq \frac{k \pi^2}{L^2},$$ 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, where the sub-Laplacian is the standard hypoelliptic operator of CR and quaternionic contact geometries, $L = \pi$ and $k=2d$ or $4d$, respectively.
This is a joint work with L. Rizzi (CMAP, École Polytechnique) and M. Seri (University College of London).

PGMO Days 2015
ENSTA ParisTech, Palaiseau, France