The Ernst equation and the Riemann-Hilbert problem on hyperelliptic Riemann surfaces (Q1377240)

The Ernst equation for a complex-valued function \(f\) of one complex variable \(z\) is given by \[ f_{z\overline z} +{1\over 2 (z+ \overline z)} (f_z+f_{\overline z})= {2 \over f+ \overline f} f_zf_{\overline z}. \] The goal of the paper under review is to show how one can construct the solutions of this equation from solutions of some generalized scalar Riemann-Hilbert problem on hyperelliptic Riemann surfaces.