Measure contraction properties of Carnot groups

By L. Rizzi. 2015

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We prove that any corank 1 Carnot group of dimension $$k+1$$k+1 equipped with a left-invariant measure satisfies the $$\mathrm {MCP}(K,N)$$MCP(K,N) if and only if $$K \le 0$$K≤0 and $$N \ge k+3$$N≥k+3. This generalizes the well known result by Juillet for the Heisenberg group $$\mathbb {H}_{k+1}$$Hk+1 to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number $$k+3$$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 $$\mathrm {MCP}(0,N)$$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, our results improve 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5eb6265b265680847995fb6901544fbdaf835df8

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Sub-Riemannian Curvature in Contact Geometry (2015, background): “6] (see also [25] for general metric measure spaces).”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, background): “, see [Oht09]. • Sub-Riemannian manifolds. The following are all examples of essentially non-branching MCP(K,N)-spaces: the (2n + 1)-dimensional Heisenberg group [Jui09], any co-rank one Carnot group [Riz16], any ideal Carnot group [Rif13], any generalized H-type Carnot group of rank k and dimension”
New formulas for the Laplacian of distance functions and applications (2018, background): “P(K,N) condition. • Sub-Riemannian manifolds. The following are all examples of essentially non-branching MCP(K,N)- spaces: the (2n+ 1)-dimensional Heisenberg group [39], any co-rank one Carnot group [50], any ideal Carnot group [49], any generalized H-type Carnot group of rank kand dimension n[8]. ”