Some Characterizations of Weakly Uniformly Perfect Sets

Abstract: In this paper, the concept of weakly uniform perfectness is considered. As an analogue of the theory of uniform perfectness, we obtain the relationships between weakly uniform perfectness and Bergman kernel, Poincar\'e metric and Hausdorff content. In particular, for a bounded domain $\Omega \subset \mathbb{C}$, we show that the uniform perfectness of $\partial \Omega$ is equivalent to $K_{\Omega}(z) \gtrsim \delta(z)^{-2}$, where $K_{\Omega}(z)$ is the Bergman kernel of $\Omega$ and $\delta(z)$ denotes the boundary distance.