Bergman functions on weakly uniformly perfect domains II

We study the boundary asymptotic behavior of Bergman functions on planar domains. We prove that $\partial\Omega$ is uniformly perfect if and only if $K_{\Omega}^{(1)}(w)\asymp \delta_{\Omega}(w)^{-4}$. For weakly uniformly perfect boundaries, we establish lower and upper bounds for $K_{\Omega}^{(1)}$, which characterize weak uniform perfectness. Combining these estimates with capacity upper bounds for the Bergman kernel, we derive optimal growth rates for the Bergman metric and Bergman distance on Zalcman-type domains; in particular, we obtain $d_{\Omega}\gtrsim \log\log |z|^{-1}$ in the power-type case and $d_{\Omega}\gtrsim (\log |z|^{-1})/(\log\log |z|^{-1})$ in the logarithmic-type case. Our results extend and refine earlier work of Chen and Xiong--Zheng.