Riemann-Hilbert Problems for the Ernst Equation and Fibre Bundles

Details Riemann-Hilbert Problems for the Ernst Equation and Fibre Bundles Riemann-Hilbert Problems for the Ernst Equation and Fibre Bundles

1998-11-12

arxiv.org/abs/gr-qc/9811039v1

J.Geom.Phys. 30 (1999) 331-342

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

12 pages, to be published in Journal of Geometry and Physics

Scientific paper

10.1016/S0393-0440(98)00067-9

Riemann--Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary value problems. The solution of a boundary value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved,it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.

Affiliated with

Also associated with

No associations

Rating

Riemann-Hilbert Problems for the Ernst Equation and Fibre Bundles does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Riemann-Hilbert Problems for the Ernst Equation and Fibre Bundles, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Riemann-Hilbert Problems for the Ernst Equation and Fibre Bundles will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-691646