Measure contraction properties of Carnot groups

Luca Rizzi (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We prove that any corank 1 Carnot group of dimension k + 1 equipped with a left-invariant measure satisfies the MCP(K, N) iff K ≤ 0 and N ≥ k + 3. This generalizes the well known result by Juillet for the Heisenberg group H 2d+1 to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number k + 3 coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least N such that the MCP(0, N) is satisfied). We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, this result improves a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat.

Mots clés: sub-riemannian; carnot groups; measure contraction property; geodesic dimension; metric measure spaces

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Optimisation et contrôle [math.OC]

Luca Rizzi. Measure contraction properties of Carnot groups. 2016. ⟨hal-01218376v2⟩