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 Hk+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.
Measure contraction properties of Carnot groups / Rizzi, L.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 55:(2016), pp. 1-20. [10.1007/s00526-016-1002-y]
Measure contraction properties of Carnot groups
Rizzi, L.
2016-01-01
Abstract
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 Hk+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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.