The Quasi Curvature-Dimension Condition with applications to\n sub-Riemannian manifolds

2019/08/05 by Emanuel Milman, Milman, Emanuel · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1908.01513

openalex publication_date 2019/08/05 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

We obtain the best known quantitative estimates for the L^p-Poincar 'e and\nlog-Sobolev inequalities on domains in various sub-Riemannian manifolds,\nincluding ideal Carnot groups and in particular ideal generalized H-type Carnot\ngroups and the Heisenberg groups, corank 1 Carnot groups, the Grushin plane,\nand various H-type foliations, Sasakian and 3-Sasakian manifolds. Moreover,\nthis constitutes the first time that a quantitative estimate independent of the\ndimension is established on these spaces. For instance, the Li-Yau / Zhong-Yang\nspectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up\nto a factor of 4.\n We achieve this by introducing a quasi-convex relaxation of the\nLott-Sturm-Villani \CD(K,N) condition we call the Quasi\nCurvature-Dimension condition \QCD(Q,K,N). Our motivation stems from\na recent interpolation inequality along Wasserstein geodesics in the ideal\nsub-Riemannian setting due to Barilari and Rizzi. We show that on an ideal\nsub-Riemannian manifold of dimension n, the Measure Contraction Property\n\MCP(K,N) implies \QCD(Q,K,N) with Q = 2^N-n \≥ 1,\nthereby verifying the latter property on the aforementioned ideal spaces; a\nresult of Balogh-Krist 'aly-Sipos is used instead to handle non-ideal corank\n1 Carnot groups. By extending the localization paradigm to completely general\ninterpolation inequalities, we reduce the study of various analytic and\ngeometric inequalities on \QCD spaces to the one-dimensional case.\nConsequently, we deduce that while (strictly) sub-Riemannian manifolds do not\nsatisfy any type of \CD condition, many of them satisfy numerous\nfunctional inequalities with \exactly the same quantitative dependence\n(up to a factor of Q) as their \CD counterparts.\n

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• Universal non-CD of sub-Riemannian manifolds

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