Curvature exponent of sub-Finsler Heisenberg groups

TL;DR

This paper investigates the curvature exponent of sub-Finsler Heisenberg groups, establishing lower bounds and characterizing when equality occurs, revealing the relationship between sub-Finsler and sub-Riemannian geometries.

Contribution

It proves that the curvature exponent is at least 5 for sub-Finsler Heisenberg groups and characterizes the case of equality, also showing the flexibility of the exponent for different structures.

Findings

Curvature exponent $N_{curv} \\geq 5$ for sub-Finsler Heisenberg groups.

Equality $N_{curv} = 5$ iff the structure is sub-Riemannian.

Existence of sub-Finsler structures with arbitrary $N \\geq 5$.

Abstract

The curvature exponent $N_{\mathrm{curv}}$ of a metric measure space is the smallest number $N$ for which the measure contraction property $\mathsf{MCP}(0,N)$ holds. In this paper, we study the curvature exponent of sub-Finsler Heisenberg groups equipped with the Lebesgue measure. We prove that $N_{\mathrm{curv}} \geq 5$, and the equality holds if and only if the corresponding sub-Finsler Heisenberg group is actually sub-Riemannian. Furthermore, we show that for every $N\geq 5$, there is a sub-Finsler structure on the Heisenberg group such that $N_{\mathrm{curv}}=N$.