Elliptic capacity and its distortion under conformal mapping
For two points \(a,b\) in the extended complex plane \(\widehat{\mathbb C}\) let \[ [a,b]=\frac{\text{abs}{a-b}}{\text{abs}{1+\overline{a}b}}; \text{ (if } b=\infty, [a,\infty]=\text{abs}{a}^{-1}). \] This is the pseudoelliptic distance of \(a,b\); it does not satisfy the triangle inequality. If \(a\in \widehat{\mathbb C}\) and \(E\subset \widehat{\mathbb C}\), we define the antipodal point \(a^*\) and the antipodal set \(E^*\) by \(a^*=-1/\overline{a}\) and \(E^*=\{a^*:a\in E\}\). \textit{R. Kühnau} [ J. Reine Angew. Math. 234, 216-220 (1969; Zbl 0185.32802)], following ideas of M.Tsuji, developed a theory of elliptic transfinite diameter and elliptic capacity. In the paper under review the authors continue this development. They give an extremal length description of the elliptic capacity \(d_e(E)\) of \(E\subset \widehat{\mathbb C}\): \(d_e(E)=\exp (-\pi\lambda(C,C^*))\), where \(\lambda(C,C^*)\) is the extremal distance between the outer boundaries \(C, C^*\) of \(E,E^*\). They also develop a theory of elliptic Robin capacity in analogy with the theories of Robin and hyperbolic Robin capacities (see e.g. the articles by \textit{P. Duren} [Proceedings of the 3rd CMFT conference on computational methods and function theory 1997, Nicosia, Cyprus, October 13-17, 1997. Singapore: World Scientific. Ser. Approx. Decompos. 11, 177-190 (1999; Zbl 0939.30016)] and by \textit{P. Duren} and \textit{J. Pfaltzgraff} [J. Anal. Math. 78, 205-218 (1999; Zbl 0974.31001)]). Finally, they use a variational method to characterize the elliptic Robin capacity by an extremal property of conformal mappings.
scientific article; zbMATH DE number 1997293