Research on the Group Structure of the Einstein Equations Solution Manifold

由于真空和电真空爱因斯坦方程的解越来越复杂，使用基于Belinskiii-Zakharov，Hoenselaers-Kinnersley-Xanthopoulos，Alekseev，Sibgatullin和Hauser-Ernst的一般形式主义的程序生成，很少通过直接代入爱因斯坦方程来检查，因此验证这些形式主义的理论基础并确定其任何局限性变得尤为重要。 恩斯特博士和豪瑟博士将通过建立各种形式主义之间的数学关系，并试图在一个综合的框架内，对各种形式主义最终必须基于的所有默认或尚未证明的假设提供明确的陈述和真正的证明，来阐明解生成机制的逻辑基础。 这包括，但不限于，一个仔细规范域的功能和可微性的要求，证明等价的各种积分方程和相关的黎曼-希尔伯特问题，证明这些方程的解决方案的存在性，并证明这些解决方案满足爱因斯坦方程。

Because solutions of ever-greater complexity of the vacuum and electrovac Einstein equations, generated using procedures based upon general formalisms of Belinskii- Zakharov, Hoenselaers-Kinnersley-Xanthopoulos, Alekseev, Sibgatullin, and Hauser-Ernst, are rarely checked by direct substitution into the Einstein equations, it has become especially important to validate the theoretical basis for and to identify any limitations of such formalisms. Dr. Ernst and Dr. Hauser will clarify the logical foundations of solution-generating machinery by establishing the mathematical relationships among the various formalisms, and by attempting to provide, within an integrated framework, clear statements and bona-fide proofs of all the tacit or as yet unproven assumptions upon which the diverse formalisms must ultimately be based. This includes, but is not limited to, a careful specification of domains of functions and differentiability requirements, proofs of equivalence of various integral equations and associated Riemann-Hilbert problems, proofs of existence of solutions of these equations, and proofs that these solutions satisfy the Einstein equations.