Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler ℓ^p-Heisenberg groups

2023/05/26 by Samuël Borza, Borza, Samuël, Kenshiro Tashiro +1
Physics and Astronomy · #26A33 #49N60 #49Q22 #53C17 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2305.16722

openalex publication_date 2023/05/26 · openalex created_date 2023/05/30 · openalex updated_date 2026/07/30

Abstract

We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property MCP(K, N), and the geodesic dimension on the Heisenberg group equipped with an ℓ^p-sub-Finsler norm. We show that for p∈(2,∞], the ℓ^p-Heisenberg group fails to satisfy any of the measure contraction properties. On the other hand, if p∈(1,2), then it satisfies the measure contraction property MCP(K, N) if and only if K ≤ 0 and N ≥ N_p, where the curvature exponent N_p is strictly greater than 2q+1 (q being the Hölder conjugate of p). We also prove that the geodesic dimension of the ℓ^p-Heisenberg group is min(2q+2,5) for p∈[1,∞). As a consequence, we provide the first example of a metric measure space where there is a gap between the curvature exponent and the geodesic dimension.

Related

• Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups
• Curvature exponent of sub-Finsler Heisenberg groups
• Failure of the curvature-dimension condition in sub-Finsler manifolds
• Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group
• Infinite geodesics of sub-Finsler distances in the Heisenberg groups