Bergman functions on weakly uniformly perfect domains II

Zhiyuan Zheng

complex analysis

Bergman functions on weakly uniformly perfect domains II

arXiv:2607.14980

summary

The paper analyzes how Bergman functions behave near the boundaries of planar domains, showing that uniform perfectness of the boundary is equivalent to a specific growth rate of the Bergman kernel, and provides bounds and optimal growth estimates for the Bergman metric and distance on weakly uniformly perfect and Zalcman-type domains.

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 $\partial Ω$ is uniformly perfect if and only if $K_{Ω}^{(1)} (w) ≍ δ_{Ω} (w)^{- 4}$ . We also find that under suitable weak uniform perfectness conditions, there exist sequences of points along which $b_{Ω} (w_{n}) = o (δ_{Ω} (w_{n})^{- 1})$ , providing partial evidence toward an affirmative answer. Our method relies on sharp lower and upper bounds for $K_{Ω}^{(1)}$ and $K_{Ω}$ . As an application, we obtain corresponding lower bounds for the Bergman distance on certain planar domains.

Topics & keywords

#bergman kernel #uniformly perfect domains #boundary asymptotics #planar domains #bergman metricBergman functionuniform perfectnessweakly uniformly perfectcapacity estimatesZalcman-type domainasymptotic growth