Sobolev, Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality

Abstract

By adapting some ideas of M. Ledoux [16], [17] and [19] to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in [3]. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is non negative.

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

“…In the Riemannian case, as noted by Croke, we attain equality in (9) when M is the hemisphere of the Riemannian round sphere. We prove the following extension to the sub-Riemannian setting.…”

“…Remark 2. In (9), L cannot be replaced by the sub-Riemannian diameter, as M might contain very long (non-minimizing) geodesics, for example closed ones, and L = +∞. See Appendix B for more details.…”

“…ϑ = 1). If, furthermore, every such geodesic minimizes distance up to the point of intersection with the boundary (i.e.θ = 1), then we have equality in (9) if and only if M is an hemisphere of the round sphere. See also [21] for a more general rigidity result.…”

Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

“…In the work [11] the authors proved that on a sub-Riemannian manifold with transverse symmetries, assuming natural geometric conditions, the sub-Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension estimate implies Li-Yau inequalities for positive solutions of the heat equation [9,11], Gaussian lower and upper bounds for the subelliptic heat kernel [9,11], log-Sobolev and isoperimetric inequalities [8,13], volume and distance comparison estimates [10] and a Bonnet-Myers type theorem [11]. Recently, it has been pointed out by Elworthy [20] that sub-Riemannian manifolds with transverse symmetries can be seen as Riemannian manifolds with bundle like metrics which are foliated by totally geodesic leaves.…”

The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

“…using a unique continuation argument by showing that on a qc manifold with n > 1, the "horizontal Hessian equation" implies that f satisfies an elliptic partial differential equation, Finally, a comparison of the metrics on H show the desired homothety. We should mention that an alternative to the use of the qc-conformal curvature tensor in Step 2 was found in [18] where once the isometry with the round sphere is established the authors invoke the classification of Riemannian submersions with totally geodesic fibers of the sphere due to Escobales [82].…”