Curvature exponent of sub-Finsler Heisenberg groups

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