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 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 λ1(M ) of the sub-Laplacian, λ1(M ) ≥ kπ 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 QC geometries, L = π and k = 2d or 4d, respectively.1 Abnormal geodesics are allowed, but strictly abnormal ones, not given by the Hamiltonian flow on the cotangent bundle, do not play any role in our construction. 2 We stress that Carnot groups are not Riemannian foliations if their step is > 2.

A minimal partition problem with trace constraint in the Grushin plane

“…. , X r } on a n-dimensional manifold, n ≥ r. The theory of perimeters in such spaces has been developed starting from the 1990s in [4,12,18], and isoperimetric inequalities in Carnot-Carathéodory spaces are a current object of investigation, see [26,25,18,9,10]. Given α ≥ 0, the Grushin plane is defined endowing R 2 with the family of vector fields X α = {∂ x , |x| α ∂ y }, where (x, y) denotes a point in R 2 and ∂ x , ∂ y respectively denote the partial derivative with respect to the first and to the second coordinate.…”

Hardy-Type Inequalities for the Carnot–Carathéodory Distance in the Heisenberg Group

“…By definition, it holds thatγ θ,pz (t) = ∇ H δ(γ θ,pz (t)) for t ∈ (0, 2π/p z ), and one could use (22) to define polar coordinates Ψ :…”

“…This section is devoted to the proof of the following result on the full Hardy constant c n (Σ), defined in (15). Its proof is based on a consequence of the sub-Riemannian Santaló formula, presented in [22].…”

“…Indeed, m(p) is the maximum of the quantity L appearing in [22,Proposition 2]. Observe that [22, Proposition 2] is valid under a compactness assumption on the ambient space.…”

An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

“…By the similar technique, one of the authors also show the containment problem for the geometric probability in H 1 [13]. Recently Prandi, Rizzi, Seri [20] show a sub-Riemannian version of the classical Santaló formula which is applied to finding the lower bound of sub-Laplacian in a compact domain with boundary. The other approaches can be referred to [19] (sub-Riemannian), and [18] (Carnot groups).…”