An example of an open Riemann surface not uniformly large with respect to Green's functions
In the present paper the author compares two properties, called (G) and (H), respectively, which would reflect certain uniformity of Riemann surfaces. A Riemann surface R is said to have the property (G) if it has the Green function g(p,q) and there exists a constant \(M>0\) such that \(\{\) \(p\in R:\) \(g(p,q)>M\}\) is simply connected for every \(q\in R\). R is said to have the property (H) if there is a constant \(\epsilon >0\) such that \(\{\) \(p\in R:\) \(d_ R(p,q)<\epsilon \}\) is simply connected for every \(q\in R\), where \(d_ R\) denotes the hyperbolic distance on R. These two properties have appeared in the study of BMO for Riemann surfaces. It is seen that (G) implies (H). The main objective of the paper under review is to prove that there exists a regular Riemann surface of Parreau-Widom type which satisfies (H) but not (G). In order to construct a surface of this property, the author takes two kinds of finite bordered surfaces. The one is a compact bordered surface \(R_ 1\) of genus one and with a single contour and its Nielsen kernel is denoted by \(S_ 1\), so that its contour \(d_ 1\) is a geodesic, whose length is denoted by c. The other is a compact bordered surface \(S_ 0\) of genus one and with two contours which are geodesics of the same length c. Let \(W=R_ 1-(S_ 1-d_ 1)\). Roughly speaking, one then constructs finite bordered surfaces \(S'_ j\) by pasting together \(S_ 1\), j-1 copies of \(S_ 0\), and W sequentially in this order. Then, a surface R having the desired property is obtained by pasting \(S'_ 1,S'_ 2,S'_ 3,..\). sequentially in this order along certain short geodesics. See the original paper for technical details.
scientific article; zbMATH DE number 4054012