Blaschke product for bordered surface

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.