标题： 单值化变换方法在广义相对论中厄恩斯坦方程的求解 显示英文标题

标题： Monodromy transform approach to solution of the Ernst equations in General Relativity

摘要： 这种方法称为“单色变换”，为解决所有已知的爱因斯坦方程在纯真空引力场情况下，存在引力相互作用的无质量场的情况下，以及某些弦理论诱导的引力模型的空间-时间对称约化提供了通用基础。在本通信中，我们介绍了该方法的关键点，应用于真空的爱因斯坦方程以及可约化为已知恩斯特方程的电真空场的爱因斯坦-麦克斯韦方程。还讨论了单色数据的定义，单色变换的直接和逆问题的表述与求解，其解的存在性和唯一性的证明，基本线性奇异积分方程的结构及其正则化，这些正则化导致(quasi-)弗雷德霍姆类型的方程。还以一致收敛的函数级数形式给出了这些方程的一般局部解的构造。 显示英文摘要

摘要： The approach, referred to as "monodromy transform", provides some general base for solution of all known integrable space - time symmetry reductions of Einstein equations for the case of pure vacuum gravitational fields, in the presence of gravitationally interacting massless fields, as well as for some string theory induced gravity models. In this communication we present the key points of this approach, applied to Einstein equations for vacuum and to Einstein - Maxwell equations for electrovacuum fields in the cases, reducible to the known Ernst equations. Definition of the monodromy data, formulation and solution of the direct and inverse problems of the monodromy transform, a proof of existence and uniqueness of their solutions, the structure of the basic linear singular integral equations and their regularizations, which lead to the equations of (quasi-)Fredholm type are also discussed. A construction of general local solution of these equations is given in terms of homogeneously convergent functional series.