Meromorphic functions on compact Riemann surfaces

Meromorphic Functions on Certain Riemann Surfaces

1. Throughout the paper we shall denote by R a Riemann surface. For a domain Í2 in P, we represent by AB(Q) the class of all the singlevalued bounded analytic functions on the closure Ü. For a meromorphic function / on a domain ß, we use the notation viw\f, Q.) to express the number of times that/ attains w in ß. Definition 1. We say that REWIb if the maximum principle suplen \fip)\ =sup3,ean \...

Meromorphic Differentials with Twisted Coefficients on Compact Riemann Surfaces

This note is to concern a generalization to the case of twisted coefficients of the classical theory of Abelian differentials on a compact Riemann surface. We apply the Dirichlet’s principle to a modified energy functional to show the existence of differentials with twisted coefficients of the second and third kinds under a suitable assumption on residues. 1 Main results and discussion Let X be...

Notes on Compact Riemann Surfaces

Non-abelian vortices on compact Riemann surfaces

We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field φ with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface Σ, their moduli space of solutions is closely related to a moduli space of τ -stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix φ, we show t...

The resultant on compact Riemann surfaces

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the ...