Journal of Differential Geometry

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

Dario Prandi, Luca Rizzi, and Marcello Seri

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper we prove 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. 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-Riemannian 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,

\[ \lambda_1 (M) \geq \frac{k \pi^2}{L^2} \; \textrm{,} \]

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:

\[ \mathbb{S}^1 \hookrightarrow \mathbb{S}^{2d+1} \overset{p}{\to} \mathbb{CP}^d \; \textrm{,} \qquad \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3} \overset{p}{\to} \mathbb{HP}^d \; \textrm{,} \qquad {d \geq 1} \; \textrm{,} \]

where the sub-Laplacian is the standard hypoelliptic operator of CR and QC geometries, $L = \pi$ and $k = 2d$ or $4d$, respectively.

Article information

Source
J. Differential Geom., Volume 111, Number 2 (2019), 339-379.

Dates
Received: 7 December 2015
First available in Project Euclid: 6 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1549422105

Digital Object Identifier
doi:10.4310/jdg/1549422105

Mathematical Reviews number (MathSciNet)
MR3909911

Zentralblatt MATH identifier
07015573

Citation

Prandi, Dario; Rizzi, Luca; Seri, Marcello. A sub-Riemannian Santaló formula with applications to isoperimetric inequalities and first Dirichlet eigenvalue of hypoelliptic operators. J. Differential Geom. 111 (2019), no. 2, 339--379. doi:10.4310/jdg/1549422105. https://projecteuclid.org/euclid.jdg/1549422105


Export citation