Ring domains with separating circles or separating annuli
Let R be a ring domain on the Riemann sphere \({\mathbb{P}}\) with complementary components E and F. Suppose \(\infty \in F\) whenever R is acutally contained in the complex plane \({\mathbb{C}}\). Two problems for ring domains are considered. First, when is there a circle in R that separates E and F? The second problem only makes sense for ring domains in \({\mathbb{C}}:\) When a separating circle exists, determine a lower bound on the size of the largest separating annulus. Answers to these problems are given in terms of two measures of the size of R - the conformal modulus mod(R) of R and the Euclidean modulus \[ e-mod(R)=(1/2\pi)\log (1+2\rho (R)),\quad where\quad \rho (R)=dist(E,F)/diam(E). \] The euclidean modulus is normalized so it agrees with the conformal modulus on annuli. The existence of separating circles is considered in three different situations. If \(0\in E\), then \(mod(R)>1/2\) is best possible for the existence of a separating circle centered at the origin. This is due to Teichmüller. For a general ring domain on \({\mathbb{P}}\), \(mod(R)>1/4\) is the sharp lower bound for the existence of a separating circle. (This result was obtained independently by \textit{A. Yu. Solynin}, Separation of continua by circles, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Vol. 168, 154-157 (1988; Zbl 0691.30015).) For the existence of a separating circle concentric with the circumcircle about E when R is contained in \({\mathbb{C}}\), \(mod(R)>0.36..\). is best possible. Sharp analogs for all three results are given in terms of the euclidean modulus. The existence of separating annuli is considered just in the first situations of separating circles. Sharp answers are given in both cases relative to the euclidean modulus, while a sharp answer in terms of the conformal modulus is obtained only in the first. Roughly speaking, a ring domain in \({\mathbb{C}}\) contains a separating annulus with modulus nearly equal to that of the ring domain. More precisely, if \(mod(R)>0.461\), then R contains a separating annulus A with \(mod(A)>mod(R)-0.461\). One consequence is that the conformal and euclidean modulus of a ring domain in \({\mathbb{C}}\) are almost equal; their difference is bounded by an absolute constant \(K\leq 0.461\).
scientific article; zbMATH DE number 4142396