Published Paper

Inserted: 10 oct 2024
Last Updated: 18 jul 2025

Journal: SIAM Journal on Mathematical Analysis
Volume: 57
Number: 4
Pages: 3561-3586
Year: 2025
Doi: 10.1137/24M1690692
Links: Journal URL, arXiv eprint

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 $N_{\mathrm{curv}}=N$.

Keywords: Heisenberg group, sub-Finsler geometry, nonsmooth analysis