Riemann-Hilbert Problems for the Ernst Equation and Fibre Bundles

TL;DR

This paper explores Riemann-Hilbert problems related to the Ernst equation using fibre bundles, enabling a unified approach to both compact and non-compact cases and facilitating explicit solutions via hyperelliptic theta functions.

Contribution

It introduces a fibre bundle framework for Riemann-Hilbert problems on Riemann surfaces, unifying treatment of different cases and applying general existence theorems.

Findings

Unified treatment of compact and non-compact Riemann surfaces.

Explicit solutions in terms of hyperelliptic theta functions for rational jump data.

Application of general existence theorems to Riemann-Hilbert problems.

Abstract

Riemann--Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary value problems. The solution of a boundary value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved,it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.