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• W2740574012 abstract "We prove a sub-Riemannian version of the classical Santalo 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 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 ∂M and negligible characteristic set. Moreover, we prove a universal (i.e. curvature independent) lower bound for the first Dirichlet eigenvalue λ1(M) of the intrinsic sub-Laplacian, λ1(M) ≥ kπ2 L2 , in terms of the rank k of the distribution and the length L of the longest reduced subRiemannian geodesic contained in M . All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations: S ↪→ S p −→ CP, S ↪→ S p −→ HP, d ≥ 1, where the sub-Laplacian is the standard hypoelliptic operator of CR and quaternionic contact geometries, L = π and k = 2d or 4d, respectively." @default.
• W2740574012 created "2017-08-08" @default.
• W2740574012 creator A5035358087 @default.
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• W2740574012 date "2015-09-17" @default.
• W2740574012 modified "2023-10-16" @default.
• W2740574012 title "A sub-Riemannian Santal'o formula with applications to isoperimetric inequalities and Dirichlet spectral gap of hypoelliptic operators" @default.
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