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

Sub-Riemannian geometry is a generalization of Riemannian one, to include nonholonomic constraints. In this paper we prove a nonholonomic 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 assumptions, 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 $\lambda_1(M)$ of the intrinsic sub-Laplacian, \begin{equation} \lambda_1(M) \geq \frac{k \pi^2}{L^2}, \end{equation} 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: \begin{equation} \mathbb{S}^1\hookrightarrow \mathbb{S}^{2d+1} \xrightarrow{p} \mathbb{CP}^d, \qquad \mathbb{S}^3\hookrightarrow \mathbb{S}^{4d+3} \xrightarrow{p} \mathbb{HP}^d, \qquad d \geq 1, \end{equation} where the sub-Laplacian is the standard hypoelliptic operator of CR and quaternionic contact geometries, $L = \pi$ and $k=2d$ or $4d$, respectively.