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1997 | 41 | 1 | 195-204
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Algebro-geometric approach to the Ernst equation I. Mathematical Preliminaries

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1. Introduction. It is well known that methods of algebraic geometry and, in particular, Riemann surface techniques are well suited for the solution of nonlinear integrable equations. For instance, for nonlinear evolution equations, so called 'finite gap' solutions have been found by the help of these methods. In 1989 Korotkin [9] succeeded in applying these techniques to the Ernst equation, which is equivalent to Einstein's vacuum equation for axisymmetric stationary fields. But, the Ernst equation is not an evolution equation and, due to this fact, one is in this case usually confronted with boundary value problems which have not been considered there. On the other hand, Neugebauer and Meinel [10] were able to transform the boundary value problem for the rigidly rotating disk of dust into a scalar Riemann-Hilbert problem on a hyperelliptic Riemann surface and gave the solution to this problem in terms of theta functions. The methods they used were suited to the particular problem and one may ask to which extent algebro-geometric methods are useful for the solution of boundary value problems of the Ernst equation. In order to tackle this problem one should at first develop the Riemann-Hilbert technique on Riemann surfaces in detail and then apply this method in order to find solutions to the Ernst equation. The first of these two papers is devoted to the brief introduction into Riemann surface techniques (for a more detailed exposition see the cited literature). The second paper shows how the developed methods apply to the Ernst equation.
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  • Max-Planck-Society, Research Unit 'Theory of Gravitation' at the FSU Jena, Max-Wien-Platz 1, 07743 Jena, Germany
  • Max-Planck-Society, Research Unit 'Theory of Gravitation' at the FSU Jena, Max-Wien-Platz 1, 07743 Jena, Germany
  • [1] Belokolos, E. D., Bobenko, A. I., Enol'skii, V.Z., Its, A. R. and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics, Springer, Berlin, 1994.
  • [2] B. A. Dubrovin, Sov. Math. Survey 36, (1981), 11.
  • [3] Dubrovin, B. A., Fomenko, A. T. and S. P. Novikov, Modern Geometry - Methods and Applications, Part II. The Geometry and Topology of Manifolds, Graduate Texts in Mathematics, Vol. 104, Springer, Berlin, 1985.
  • [4] H. M. Farkas and I. Kra, Riemann Surfaces, Graduate Texts in Mathematics, Vol. 71, Springer, Berlin 1993.
  • [5] John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, vol. 352, Springer, New York, 1973.
  • [6] P. A. Griffiths, Ann. of Math., (2) 90, (1969), 460.
  • [7] C. Klein and O. Richter, submitted to C. R. Acad. Sci.
  • [8] L. Koenigsberger, Vorlesungen über die Theorie der hyperelliptischen Integrale, Teubner, Leipzig, 1878.
  • [9] D. A. Korotkin, Theor. Math. Phys. 77, (1989), 1018.
  • [10] G. Neugebauer and R. Meinel, Phys. Rev. Lett, 73, (1994), 2166.
  • [11] H. Stahl, Theorie der Abelschen Funktionen, Teubner, Leipzig, 1896.
  • [12] K. Weierstrass, Vorlesungen über die Theorie der Abelschen Transcendenten, Math. Werke, Vol. 4, Berlin, 1902.
  • [13] S. I. Zverovich, Russ. Math. Survey 26, (1971), 117.
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