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We show that the class of hyperelliptic solutions to the Ernst equation (thestationary axisymmetric Einstein equations in vacuum) previously discovered byKorotkin and Neugebauer and Meinel can be derived via Riemann-Hilberttechniques. The present paper extends the discussion of the physical propertiesof these solutions that was begun in a Physical Review Letter, and suppliescomplete proofs. We identify a physically interesting subclass where the Ernstpotential is everywhere regular except at a closed surface which might beidentified with the surface of a body of revolution. The correspondingspacetimes are asymptotically flat and equatorially symmetric. This suggeststhat they could describe the exterior of an isolated body, for instance arelativistic star or a galaxy. Within this class, one has the freedom tospecify a real function and a set of complex parameters which can possibly beused to solve certain boundary value problems for the Ernst equation. Thesolutions can have ergoregions, a Minkowskian limit and an ultrarelativisticlimit where the metric approaches the extreme Kerr solution. We give explicitformulae for the potential on the axis and in the equatorial plane where theexpressions simplify. Special attention is paid to the simplest non-staticsolutions (which are of genus two) to which the rigidly rotating dust diskbelongs.Comment: 32 pages, 2 figures, uses pstricks.sty, updated version (October 7,  1998), to appear in Phys. Rev.

Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces