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

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 $K\le 0$ and $N\ge N_p$, 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.

46 pages, 11 figures. v2: 50 pages, 13 figures. To appear in the Annales de l'Institut Fourier