A closed subset \(E\) of the Riemann sphere \(\widehat{\mathbb C}\) containing at least three points is called uniformly perfect if there exists a constant \(0< c< 1\) such that \[ E\cap \{z: cr<| z-a|< r\}\neq\emptyset \] for every \(a\in E\setminus \{\infty\}\) and \(0< r< \text{diam}\,E\).
Uniformly perfect sets have been investigated by Pommerenke and others, and many equivalent conditions have been noted so far and proved to be useful in geometric function theory. The present article surveys typical conditions for uniform perfectness with explicit estimates and includes some new or improved results.
The following is one of the results. Let \(M_D\) denote the supremum of the moduli of ring domains in \(D= \widehat{\mathbb C}\setminus E\) separating E, where the modulus of a ring domain is defined as the number \(m\) when it is conformally equivalent to the annulus \(\{z: 1<| z|< e^m\}\), and a ring domain \(A\) in \(D\) is said to separate \(E\) if each component of \(\widehat{\mathbb C}\setminus A\) intersects \(E\), equivalently, \(A\) is incompressible in \(D\). And let \(I_D\) be the injectivity radius of \(D\) with respect to the hyperbolic metric of \(D\) of constant curvature \(-4\), i.e., \(I_D\) is half the infimum of the hyperbolic lengths of non-trivial closed curves in \(D\). Then we have \[ 2I_D\leq \frac{\pi^2}{M_D}\leq \min\{2I_D e^{2I_D}, 2I_D^2\coth^2 i_D\}, \] which follows partially from an improvement of Maskit’s result on the comparison between hyperbolic and extremal lengths. In particular, \(E\) is unily perfect iff \(M_D<\infty\) iff \(I_D> 0\).
This paper includes also the quantitative version of a recent result of P. Järvi and M. Vuorinen [J. Lond. Math. Soc. 54, No. 3, 515–529 (1996; Zbl 0872.30014)], which characterizes uniform perfectness in terms of Hausdorff contents. As a corollary, we have a lower estimate of Hausdorff dimension: \[ \text{H-dim}\,E\geq \frac{\log 2}{\log(2e^{M_D}+1)}. \]