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Canonical Embeddings of Pairs of Arcs and Extremal Problems on Ring Domains

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Abstract

A configuration consisting of two disjoint Jordan arcs in \(\overline{\mathbb {C}}\) is canonical if each of these arcs is a hyperbolic geodesic segment in the domain on \(\overline{\mathbb {C}}\) complementary to the other arc. In this paper, we first show how recent results on canonical configurations obtained by M. Bonk and A. Eremenko follow from J. Jenkins’s theorem on extremal partitioning of Riemann surfaces into collections of ring domains. Then we discuss several other problems where similar extremal configurations appear. Among those are the following: Teichmüller’s problem on ring domains separating two pairs of points, a problem on Jenkins–Strebel quadratic differentials with four poles, a problem on the maximal winding module number, Ahlfors’ separation problem, and extremal problems on the harmonic measure. We are mainly concerned with two-dimensional space but several three-dimensional problems also will be discussed.

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  1. Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX, 79409, USA

    Alexander Yu Solynin

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  1. Alexander Yu Solynin
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Correspondence to Alexander Yu Solynin.

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Communicated by Vladimir V. Andrievskii.

In the memory of Peter Duren, a wonderful person, a caring mentor for younger colleagues, an outstanding mathematician and an excellent writer whose research papers and books inspired so many of us.

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Solynin, A.Y. Canonical Embeddings of Pairs of Arcs and Extremal Problems on Ring Domains. Comput. Methods Funct. Theory (2023). https://doi.org/10.1007/s40315-023-00491-7

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