The Boundary Value Problem for the Stationary and Axisymmetric Einstein Equations is Generically Solvable

We prove existence, uniqueness, and regularity for Dirichlet solutions of the stationary and axisymmetric Einstein equations in vacuum and in rigidly rotating ideal fluid matter for data (on a ball) whose absolute values are in a characteristic way limited by the “diameter” of the ball. These results have important consequences for the existence of exterior vacuum and interior matter solutions for rotating stars. The mathematical procedures and results have remarkable connections with a numerical solution technique for rotating stars.