Physically Realistic Solutions to the Ernst Equation on Hyperelliptic Riemann Surfaces

It is usually assumed that stars and galaxies can be described as ideal fluids in thermodynamical equilibrium, see [1]. In general relativity, the corresponding spacetimes are stationary and axisymmetric. Unfortunately, due to the complicated structure of the field equations with matter there is only little hope that axisymmetric and stationary spacetimes can be found analytically which correspond to realistic cosmic objects. In the vacuum region, however, powerful methods are known which allow to construct solutions quite explicitly. The surface of a body of revolution is a natural boundary at which the metric functions are not regular. Therefore, one is looking for solutions to the vacuum equations that are outside the surface analytic and can continuously be extended to this surface. In other words, one has to solve a boundary value problem for the vacuum equations in which the matter enters via the boundary conditions. To put the matter into the boundary conditions is, e.g., possible for two dimensional matter distributions like disks or rings...