Curvature exponent of sub-Finsler Heisenberg groups

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 $N_{\mathrm{curv}}\ge 5$, 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 $N_{\mathrm{curv}}=N$.

25 pages, 1 figure. v2: Minor corrections and clarifications. v3: Minor corrections. To appear in the SIAM Journal on Mathematical Analysis

References in corpus (1)

• Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries

Cited by in corpus (1)

• On the lower bound of the curvature exponent on step-two Carnot groups