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

Abstract : In this paper 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 assumptions, satisfied by any sub-Riemannian structure associated with a Riemannian foliation with totally geodesic leaves (e.g.\ CR and QC manifolds with symmetries), any Carnot group, and some non-equiregular structures such as the Martinet one. A key ingredient is a ``reduction procedure'' that allows to consider only a simple subset of sub-Riemann\-ian geodesics. As an application, we derive isoperimetric-type and ($p$-)Hardy-type inequalities for a compact domain $M$ with piecewise $C^{1,1}$ boundary, and a universal lower bound for the first Dirichlet eigenvalue $\lambda_1(M)$ of the 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 QC geometries, $L = \pi$ and $k=2d$ or $4d$, respectively.
Complete list of metadatas

Cited literature [40 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01221668
Contributor : Luca Rizzi <>
Submitted on : Tuesday, March 28, 2017 - 12:09:31 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:31 PM
Long-term archiving on : Thursday, June 29, 2017 - 4:22:39 PM

File

santalo-main.pdf
Files produced by the author(s)

Identifiers

Citation

Dario Prandi, Luca Rizzi, Marcello Seri. A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators. Journal of Differential Geometry, International Press, 2019, 111 (2), pp.339-379. ⟨10.4310/jdg/1549422105⟩. ⟨hal-01221668v3⟩

Share

Metrics

Record views

641

Files downloads

186