Sets of Minimal Capacity and Extremal Domains

TL;DR

This paper investigates extremal domains of minimal capacity for meromorphic functions, proving their unique existence, analyzing their structure with quadratic differentials, and illustrating their properties through examples.

Contribution

It establishes a unique existence theorem for extremal domains of minimal capacity and develops analytical tools for their characterization, including quadratic differentials and symmetry properties.

Findings

Existence and uniqueness of extremal domains proven.

Structural analysis using quadratic differentials provided.

Geometric estimates and examples illustrated key properties.

Abstract

Let f be a function meromorphic in a neighborhood of infinity. The central problem in the present investigation is to find the largest domain D \subset C to which the function f can be extended in a meromorphic and singlevalued manner. 'Large' means here that the complement C\D is minimal with respect to (logarithmic) capacity. Such extremal domains play an important role in Pad'e approximation. In the paper a unique existence theorem for extremal domains and their complementary sets of minimal capacity is proved. The topological structure of sets of minimal capacity is studied, and analytic tools for their characterization are presented; most notable are here quadratic differentials and a specific symmetry property of the Green function in the extremal domain. A local condition for the minimality of the capacity is formulated and studied. Geometric estimates for sets of minimal capacity are given. Basic ideas are illustrated by several concrete examples, which are also used in a discussion of the principal differences between the extremality problem under investigation and some classical problems from geometric function theory that possess many similarities, which for instance is the case for Chebotarev's Problem.