Formal power series solutions of the stationary axisymmetric vacuum Einstein equations (Q1113022)

The present paper is concerned with the formal power series solution of the stationary axially symmetric vacuum (SAV) Einstein equations. By using a formal Riemann-Hilbert problem we present an explicit formula representing the formal solution in terms of a certain \(\infty \times \infty\) matrix \(\xi\). The space \(V_ 0\) of all formal solutions corresponding to the SAV space-times which has the formal Ernst potentials is determined exactly. The asymptotically flat SAV space-times constitute a subset of \(V_ 0\).