Quantitative hyperconvexity and the Bergman distance

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ be a bounded hyperconvex domain admitting a negative plurisubharmonic exhaustion function $u$ such that $-u(z)\le C\exp [-\eta (\log (1/{\delta }_{\Omega }(z)))]$ near the boundary, where $\eta$ is nondecreasing with ${\lim }_{t\to +\infty }\eta (t)=+\infty$. We present a lower bound for the Bergman distance $d_{B,\Omega }(z_0,z)$ by counting the iterates of a function determined by the generalized inverse of $\eta$. The bound tends to infinity as $z\to \partial \Omega$ for every such $\eta$; moreover, it recovers several known estimates. The proof combines a one-sided localization of Green sublevel sets with boundary estimates for the Bergman kernel and a capacity--volume inequality. We also obtain an upper bound for the Bergman distance along the axis of a family of complete Hartogs domains whose fiber radii decay faster than every positive power of the boundary distance in the base disc.