Blaschke product for bordered surface

TL;DR

This paper provides an explicit description of holomorphic ramified coverings of a disc by bordered Riemann surfaces, using elliptic theta functions, with applications in complex analysis and magnetic configurations.

Contribution

It introduces a new explicit method to describe holomorphic coverings of discs by bordered Riemann surfaces, linking complex analysis and magnetic configuration models.

Findings

Explicit description of coverings using elliptic theta functions

Application to multidimensional complex analysis problems

Connection to magnetic configurations in planar magnets

Abstract

It is well known that a ramified holomorphic covering of a closed unitary disc by another such a disc is given by a finite Blaschke product. The inverse is also true. In this note we give an explicit description of holomorphic ramified coverings of a disc by other bordered Riemann surfaces. The problem of covering a disc by an annulus arises e.g. in multidimensional complex analysis; we show that it may be effectively solved in terms of elliptic theta functions. The covering of a disc by a flat domain is discussed in a monograph by Goluzin. The machinery used here strongly resembles the description of magnetic configurations in submicron planar magnets.