Reduction of Vector Boundary Value Problems on Riemann Surfaces to One-Dimensional Problems

Abstract

This article lays foundations for the theory of vector conjugation boundary value problems on a compact Riemann surface of arbitrary positive genus. The main constructions of the classical theory of vector boundary value problems on the plane are carried over to Riemann surfaces: reduction of the problem to a system of integral equations on a contour, the concepts of companion and adjoint problems, as well as their connection with the original problem, the construction of a meromorphic matrix solution. We show that each vector conjugation boundary value problem reduces to a problem with a triangular coefficient matrix, which in fact reduces the problem to a succession of one-dimensional problems. This reduction to the well-understood one-dimensional problems opens up a path towards a complete construction of the general solution of vector boundary value problems on Riemann surfaces.