The Riemann boundary value problem on closed Riemann surfaces and integrable systems

Abstract

The Riemann problem has different forms. In several areas of physics (scattering theory, integrable systems and so on) it is necessary to build an analytic function or a matrix with prescribed poles and zeros on a Riemann surface. Another, similar, problem is to construct a section of a complex vector bundle over a Riemann surface (for example, anti-self-dual Yang-Mills equation). As was established by A. Grothendieck and H. Röhrl, this problem is equivalent to the Riemann boundary value problem: let M be a compact Riemann surface, Γ a contour on M, G(p) an n × n matrix on Γ; determine an analytic matrix F(p) in MΓ which is multiple of a given divisor ∗⧹γ and satisfies the boundary condition on Γ F + ( p) = G( p) F −( p). This problem is well understood on the complex plane. It is related to singular integral operators, Wiener-Hopf equations and Banach algebras. In the case of a Riemann surface there are additional with the Riemann-Roch and Abel theorems, Jacobi varieties and Riemann theta-functions. These relations are described here. The Riemann problem is equivalent to the so-called ∂ -problem ∂u ∂ λ = Au . Recently this elliptic equation was used to study the inverse scattering problem. Finally, we consider some physical problems (finite-gap potentials, the Landau-Lifschitz equation, the Painlevé problems) to illustrate the applications of the Riemann problem on a Riemann surface. Acquaintance with Riemann surface theory and algebraic topology is not assumed.