Bergman functions on weakly uniformly perfect domains II

Abstract

We study the boundary asymptotic behavior of Bergman functions on planar domains. Motivated by Chen's question on the equivalence between uniform perfectness of the boundary and the sharp growth rates of the Bergman kernel and the Bergman metric, we focus on the second part of the question concerning the Bergman metric. We prove that $\partial\Omega$ is uniformly perfect if and only if $K_{\Omega}^{(1)}(w)\asymp \delta_{\Omega}(w)^{-4}$. We also find that under suitable weak uniform perfectness conditions, there exist sequences of points along which $b_{\Omega}(w_n)=o(\delta_{\Omega}(w_n)^{-1})$, providing partial evidence toward an affirmative answer. Our method relies on sharp lower and upper bounds for $K_{\Omega}^{(1)}$ and $K_{\Omega}$. As an application, we obtain corresponding lower bounds for the Bergman distance on certain planar domains.