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arXiv · 2407.14619

Curvature exponent of sub-Finsler Heisenberg groups

Samuël Borza·Mattia Magnabosco·Tommaso Rossi·Kenshiro Tashiro
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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 $N_{\mathrm{curv}}=N$.

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Samuël Borza, Mattia Magnabosco, Tommaso Rossi, Kenshiro Tashiro. 2024-07-19. Curvature exponent of sub-Finsler Heisenberg groups. https://doi.org/10.1137/24m1690692

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