Analytic immersions of parabolic Riemann surfaces.

Abstract

Under conformal equivalence the set of all parabolic Riemann surfaces is divided into equivalence classes. Since a conformal equivalence is an analytic immersion, then the study of analytic immersions between all pairs of parabolic Riemann surfaces reduces to the study of immersions between all equivalence classes by representatives. The complex plane, the "cylinder", and the set of all torii modulo conformal equivalence form a useful set of representatives which enables us to determine easily whether analytic immersions exist or not. Where they do exist the use of the fiber map theorem permits us to give these analytic immersions as analytic immersions between complex planes. In particular, in the case of a pair of torii, we have been able to find a necessary and sufficient criterion to determine the existence of analytic immersions. Furthermore if there exist any then we can determine all immersions.