Geometric inequalities on Heisenberg groups

Abstract

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group ${\mathbb{H}}^n$. 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 ${\mathbb{H}}^n$ 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.

to appear in Calculus of Variations and Partial Differential Equations (42 pages, 1 figure)