The stationary axisymmetric Einstein equations in vacuum are equivalent to the complex Ernst equation. The Ernst equation is completely integrable, i.e. it can be treated as the integrability condition of an overdetermined linear differential system. The spectral parameter of the system is defined on a family of Riemann surfaces of genus zero whose branch points are parametrized by the physical coordinates. To solve boundary value problems for the Ernst equation, a Riemann-Hilbert problem is formulated. In the scalar case, this problem can be solved in terms of a Cauchy integral which is equivalent to the Poisson integral in the case of disks. The corresponding solutions are static where the Ernst equation reduces to the Laplace equation. The matrix problem can be gauge transformed to a scalar problem on a Riemann surface for analytic jump data. In the case of rational jump data, the Riemann surface is compact and the corresponding solution to the Ernst equation can be given in terms of Korotkin's hyperelliptic solutions. On the axis and at the additional branch points of the surface, the Riemann surface degenerates. Using results of Fay and Yamada on can show that the solutions are however regular at these points. Similarly one can treta the static limit of the solutions. In contrast to the Poisson integral, the solutions may have singularities in addition to the jump related to the contour of the Riemann-Hilbert problem.