ASYMPTOTICALLY FLAT SOLUTIONS TO THE ERNST EQUATION WITH REFLECTION SYMMETRY

It is shown that the class of asymptotically Bat solutions to the axis ymmetric and stationary vacuum Einstein equations with reflection symm etry of the metric is uniquely characterized by a simple relation for the Ernst potential f((u)) on the upper part of the symmetry axis (zet a axis): f((u))(zeta)(f) over bar((u))(-zeta) = 1 This result generali zes a well-known fact from potential theory: axisymmetric solutions to the Laplace equation that vanish at infinity and have reflection symm etry with respect to the plane zeta = 0 are characterized by a potenti al that is an odd function of zeta on the upper part of the zeta axis.