Optimal Linear Dependence on Boundary Type for Local Gromov Hyperbolicity of the Kobayashi Metric

The Gromov hyperbolicity constant of a metric space $(X,d)$ is the infimum of all $\delta \ge 0$ such that $(X,d)$ is $\delta$-hyperbolic. For a Kobayashi hyperbolic domain $\mathrm{\Omega }\subset {\text{C}}^n$ and a boundary point $p\in \mathrm{\partial }\mathrm{\Omega }$, let ${\delta }_{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega },p)$ denote the local Gromov hyperbolicity constant obtained by restricting the points to arbitrarily small Euclidean neighborhoods of $p$, while distances are still measured by the ambient Kobayashi distance $K_{\mathrm{\Omega }}$. For each even integer $M\ge 2$, let $\mathfrak{H}(M)$ be the supremum of these constants over all complex dimensions $n\ge 2$ and all domains whose boundary is smooth and convex near the distinguished point and has D’Angelo type at most $M$ at that point. We prove $$\frac{\log 2}{2}M\le \mathfrak{H}(M)<36M\mathrm{.}$$ Thus the optimal universal dependence of the local Gromov hyperbolicity constant on boundary type is linear. We also show that no analogous bound holds for the global Gromov hyperbolicity constant, even among smooth bounded strongly convex domains.