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Sharp measure contraction property for generalized H-type Carnot groups

Institut de Mathématiques de Jussieu-Paris, Rive Gauche UMR CNRS 7586, Université Paris-Diderot, Batiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France

E-mail Address: davide.barilari@imj-prg.fr

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Luca Rizzi

CNRS, Institut Fourier, Université Grenoble Alpes, F-38000 Grenoble, France

E-mail Address: luca.rizzi@univ-grenoble-alpes.fr

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https://doi.org/10.1142/S021919971750081XCited by:21 (Source: Crossref)

Abstract

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $MCP (K, N)$ if and only if $K \leq 0$ and $N \geq k + 3 (n - k)$ . The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.

Keywords:

AMSC: 53C17, 53C22, 35R03, 54E35, 53C21

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