2024/02/22 by Borza, Samuël, Mattia Magnabosco, Tommaso Rossi +4 · 2 citations
Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2402.14779
openalex publication_date 2024/02/22 · openalex created_date 2024/02/23 · openalex updated_date 2026/07/30
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 C1 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 C1,1 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 C1 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 C1 and strongly convex norm, and with a positive smooth measure, does not satisfy the MCP(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.