Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups
Résumé
In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short MCP, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither C 1 nor strongly convex, the associated Heisenberg group does not satisfy MCP(K, N) for any pair of parameters K ∈ R and N ∈ (1, ∞). On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a C 1,1 and strongly convex norm, and with the Lebesgue measure, satisfies MCP(0, N) for some N ∈ (1, ∞). Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of C 1 and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short CD(K, N). We show that the sub-Finsler Heisenberg group, equipped with a C 1 and strongly convex norm, and with a positive smooth measure, does not satisfy the CD(K, N) condition for any pair of parameters K ∈ R and N ∈ (1, ∞). Combining this result with our findings regarding the measure contraction property, we conclude the failure of the CD condition in the Heisenberg group for every sub-Finsler structure. Contents
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