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

TL;DR

This work initiates the study of synthetic curvature-dimension bounds and geodesic dimension for sub-Finsler lp-Heisenberg groups, revealing a sharp p-dependent dichotomy. Using Pontryagin Maximum Principle and Shelupsky p-trigonometric functions, the authors derive explicit formulas for the exponential map and its Jacobian, connecting MCP(0,N) to a differential inequality on the reduced Jacobian and establishing MCP(0,N) only for p∈(1,2) with N≥N_p (where N_p>2q+1). They also compute the geodesic dimension, showing N_geo = 5 for p∈[1,3] and N_geo = 2q+2 for p∈[3,∞), thereby producing the first example of a metric measure space with a gap between the curvature exponent and geodesic dimension in a sub-Finsler setting. The boundary cases p=1 and p=∞ exhibit MCP failure and a minimal geodesic dimension of 4 due to branching and a large cut locus, underscoring substantial differences from sub-Riemannian and Finsler geometries.

Abstract

We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property $\mathsf{MCP}(K, N)$, and the geodesic dimension on the Heisenberg group equipped with an $\ell^p$-sub-Finsler norm. We show that for $p\in(2,\infty]$, the $\ell^p$-Heisenberg group fails to satisfy any of the measure contraction properties. On the other hand, if $p\in(1,2)$, then it satisfies the measure contraction property $\mathsf{MCP}(K, N)$ if and only if $K \leq 0$ and $N \geq N_p$, 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 $\ell^p$-Heisenberg group is $\min(2q+2,5)$ for $p\in[1,\infty)$. 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.

Figures (13)

• Figure 1: A summary of the main theorems of this work on the curvature exponent and the geodesic dimension of the $\ell^p$-Heisenberg groups.
• Figure 2: Geometric definition of the $p$-trigonometric functions.
• Figure 3: Graphs of the $p$-trigonometric functions $\sin_p(\theta)$ and $\cos_p(\theta)$
• Figure 4: Some geodesics of the $\ell^p$-Heisenberg group from the origin. The dotted curves represents the projections of the geodesics onto the plane, while the red points are where the geodesics lose smoothness.
• Figure 5: The projection of a geodesic $t \mapsto \gamma(t) := \mathrm{exp}^t_e(r,\theta,w)$ of the $\ell^p$-Heisenberg group is an $\ell^q$-arc. Here the red denotes the area, the blue does the geodesic, and the green does the initial vector.

Theorems & Definitions (84)

• Theorem A: Curvature exponent for $p\in(1,\infty) \setminus \{2\}$, \ref{['thm:MCPp>2']} and \ref{['thm:MCPp<2']}
• Theorem B: Geodesic dimension for $p\in(1,\infty)$, \ref{['thm:geodim']}
• Theorem C: $\mathsf{MCP}$ and geodesic dimension for $p=1,\infty$, \ref{['thm:ell1mcp']} and \ref{['thm:ell1geodim']}
• Definition 1
• Definition 2
• Definition 3: ohta2007
• Remark 4
• Proposition 5: Same proof as in sturm2006
• Definition 6
• Definition 7