Hyperbolic capacity and its distortion under conformal mapping
It is well-known that the transfinite diameter \(d(E)\) of a closed set \(E\) of (for example) curves in the complex plane \(\mathbb{C}\) is conformally invariant. The problem to minimize the transfinite diameter of a subset of th curves \(E\) is discussed by \textit{P. L. Duren} and \textit{M. M. Schiffer} [Pac. J. Math. 148, No. 2, 251-273 (1991; Zbl 0721.30018)], Complex Variables, Theory Appl. 21, No. 3-4, 189-196 (1993; Zbl 0793.30018)] and leads to the Robin function, the Green's function of the mixed boundary value problem of the Laplacian and to the Robin capacity (for more details, see the survey paper by \textit{P. L. Duren} [Computational Methods and Function Theory \textit{N. Papamichael}, \textit{St. Ruscheweyh} and \textit{E. B. Saff}, eds.), World Scientific, Singapor. 177-190 (1999; Zbl 0939.30016)]. The paper under review deals with the hyperbolic case. The concept of the hyperbolic Robin capacity is introduced. Let \(\Omega\) be a domain in the unit disk and let \(A\subset\partial \Omega\subset \Pi\) be a subset of its inner boundary, where \(\Pi\) is the unit circle, then the hyperbolic Robin capacity \(\delta_h(A)\) of \(A\) with respect to \(\Omega\) is defined as \(\delta_h(A)=\exp\{-2\pi \lambda_\Omega (A,\Pi)\}\), where \(A_\Omega(A,\Pi)\) is the extremal length of curves which connect \(A\) to \(\Pi\). The authors show that the hyperbolic Robin capacity of a subset \(A\) of the curves \(E\) is the minimum of the hpyerbolic capacity among all conformal mappings of a normalized class if the set \(A\) contains full curves. Using a variational method a characterization of the extremal domains is derived. The authors remark that a preprint by M. D. O'Neill and R. E. Thurman deals with a more general case for the set \(A\). This is also the topic of a paper by the reviewer and A. Yu. Solynin [Zapiski Nauchn. Sem. POMI 263, 49-69 (2000)] which furthermore contains a discussion of the set of extremal domains.
scientific article; zbMATH DE number 1369961
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