Abstract
We prove that any corank 1 Carnot group of dimension \(k+1\) equipped with a left-invariant measure satisfies the \(\mathrm {MCP}(K,N)\) if and only if \(K \le 0\) and \(N \ge k+3\). This generalizes the well known result by Juillet for the Heisenberg group \(\mathbb {H}_{k+1}\) 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 \(\mathrm {MCP}(0,N)\) 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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Notes
Ohta defines the measure contraction property for general length spaces, possibly with non-negligible cut loci. Under our assumptions, this simpler definition is equivalent to Ohta’s, see [16, Lemma 2.3].
A sub-Riemannian structure \((M,\mathcal {D},g)\) is fat if for all \(x \in M\) and \(X \in \mathcal {D}\), \(X(x) \ne 0\), then \(\mathcal {D}_x + [X,\mathcal {D}]_x=T_xM\). It is ideal if it is complete and does not admit non-trivial abnormal minimizers.
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Acknowledgments
I warmly thank D. Barilari for many fruitful discussions, and the anonymous referee for many useful comments. This research was supported by the ERC StG 2009 “GeCoMethods”, contract n. 239748, by the iCODE institute (research project of the Idex Paris-Saclay), and by the ANR project SRGI ANR-15-CE40-0018. This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research” and from the support to this program from EDF.
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Communicated by A. Malchiodi.
