Regular Systems on Riemann Surfaces

Abstract

In this paper, we consider the current status of the Riemann–Hilbert problem and problems closely related to it. In the global theory of differential equations with regular singular points, the Riemann– Hilbert problem is one of the central topics. Two-dimensional models in contemporary theoretical physics use fundamental facts of the theory of Riemann surfaces. One of the central problems of this theory is the Riemann–Hilbert problem and questions related to it, for example, the linear conjugation problem. A natural language for the investigation of the Riemann–Hilbert problem is the language of holomorphic bundles with connections on Riemann surfaces. It makes clear the relationship with the linear conjugation problem, which is usually considered for Holder-class functions. We develop a different approach; namely, we consider the Riemann–Hilbert problem for the Carleman– Bers–Vekua system, replacing the Wiener–Hopf factorization of a matrix-valued function by the Φ-factorization. To study deformations of complex structures on Riemann surfaces, we also consider the Beltrami equation as a particular case of the Carleman–Bers–Vekua equation. We also discuss the algebraic formulation of the Riemann–Hilbert problem in the framework of the differential Galois theory, where this problem is known as the inverse problem and is interesting for us because of its constructive nature.