Functional Inequalities on sub-Riemannian manifolds via QCD

Abstract: We are interested in obtaining Poincare and log-Sobolev inequalities on domains in sub-Riemannian manifolds (equipped with their natural sub-Riemannian metric and volume measure).

It is well-known that strictly sub-Riemannian manifolds do not satisfy any type of Curvature-Dimension condition CD(K,N), introduced by Lott-Sturm-Villani some 15 years ago, so we must follow a different path. We show that while ideal (strictly) sub-Riemannian manifolds do not satisfy any type of CD condition, they do satisfy a quasi-convex relaxation thereof, which we name QCD(Q,K,N). As a consequence, these spaces satisfy numerous functional inequalities with exactly the same quantitative dependence (up to a factor of Q) as their CD counterparts. We achieve this by extending the localization paradigm to completely general interpolation inequalities, and a one-dimensional comparison of QCD densities with their "CD upper envelope". We thus obtain the best known quantitative estimates for (say) the L^p-Poincare and log-Sobolev inequalities on domains in the ideal sub-Riemannian setting, which in particular are independent of the topological dimension. For instance, the classical Li-Yau / Zhong-Yang spectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4.

No prior knowledge will be assumed, and we will (hopefully) explain all of the above notions during the talk.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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