Measure contraction property and curvature-dimension condition on sub-Finsler Heisenberg groups

In this paper, we investigate the validity of synthetic curvature-dimension bounds in the sub-Finsler Heisenberg group, equipped with a positive smooth measure. Firstly, we study the measure contraction property, in short $\mathsf{MCP}$, proving that its validity depends on the norm generating the sub-Finsler structure. Indeed, we show that, if it is neither $C^1$ nor strongly convex, the associated Heisenberg group does not satisfy $\mathsf{MCP}(K,N)$ for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. On the contrary, we prove that the sub-Finsler Heisenberg group, equipped with a $C^{1,1}$ and strongly convex norm, and with the Lebesgue measure, satisfies $\mathsf{MCP}(0,N)$ for some $N \in (1,\infty)$. Additionally, we provide a lower bound on the optimal dimensional parameter, and we also study the case of $C^1$ and strongly convex norms. Secondly, we address the validity of the curvature-dimension condition pioneered by Sturm and Lott-Villani, in short $\mathsf{CD}(K,N)$. We show that the sub-Finsler Heisenberg group, equipped with a $C^1$ and strongly convex norm, and with a positive smooth measure, does not satisfy the $\mathsf{MCP}(K,N)$ condition for any pair of parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$. Combining this result with our findings regarding the measure contraction property, we conclude the failure of the $\mathsf{CD}$ condition in the Heisenberg group for every sub-Finsler structure.