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Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2016

Measure contraction properties of Carnot groups

Résumé

We prove that any corank 1 Carnot group of dimension k + 1 equipped with a left-invariant measure satisfies the MCP(K, N) if and only if K ≤ 0 and N ≥ k + 3. This generalizes the well known result by Juillet for the Heisenberg group H k+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, 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.
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Dates et versions

hal-01218376 , version 1 (21-10-2015)
hal-01218376 , version 2 (23-02-2016)
hal-01218376 , version 3 (21-05-2016)

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Luca Rizzi. Measure contraction properties of Carnot groups . Calculus of Variations and Partial Differential Equations, 2016, ⟨10.1007/s00526-016-1002-y⟩. ⟨hal-01218376v3⟩
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