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

TL;DR

This paper explores synthetic curvature-dimension bounds in sub-Finsler geometry on the Heisenberg group with $ extit{ ext{ell}}^p$ norms, revealing conditions under which measure contraction properties hold and identifying a gap between curvature exponent and geodesic dimension.

Contribution

It introduces the first analysis of measure contraction properties and geodesic dimension in sub-Finsler $ extit{ ext{ell}}^p$-Heisenberg groups, highlighting new phenomena in metric measure spaces.

Findings

For $p o ext{infinity}$, the space fails to satisfy MCP.

For $p o 1$, MCP holds iff $K o 0$ and $N o N_p$.

Geodesic dimension is $ ext{min}(2q+2,5)$ for all $p$.

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 $K \leq 0$ and $N \geq N_p$, where the curvature exponent $N_p$ is strictly greater than $2q+1$ ($q$ being the H\"older 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.