Geometric inequalities on Heisenberg groups

2016/05/22 by Balogh, Zoltán M., Kristály, Alexandru, Sipos, Kinga · 2 citations
#49Q20 #53C17 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1605.06839

Abstract

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group \mathbb Hⁿ. Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott-Villani and Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschläger. The latter statement implies sub-Riemannian versions of the geodesic Prékopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of \mathbb Hⁿ developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.

Cited by

• Functional Liftings of Restricted Geometric Inequalities
• Universal non-CD of sub-Riemannian manifolds

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