Infinitesimally thin disks of pressureless matter, so called dust, are discussed in astrophysics as models for certain galaxies and the matter in accretion disks around black holes. Since the vacuum Einstein equations in the stationary axisymmetric case are equivalent to the completely integrable Ernst equation, global spacetimes can be constructed for these models. The matter in the disk leads to a boundary value problem for the Ernst equation which can be treated with Riemann-Hilbert techniques. In the scalar case this leads to the Poisson integral. The matrix Riemann-Hilbert problem can be gauge transformed to a scalar problem on a Riemann surface. In the case of rational jump data, this surface is compact and the corresponding solutions to the Ernst equation form a subclass of Korotkin's hyperelliptic solutions. Within this class one can study which boundary value problems can be solved on a given Riemann surface. As an example we discuss a family of disks made up of two counterrotating dust components. The complete metric is given explicitly in terms of hyperelliptic functions which are evaluated numerically.