紧致流形上Einstein约束方程的非CMC解
Non-CMC Solutions of the Einstein Constraint Equations on Compact Manifolds with Apparent Horizon Boundaries
摘要：
在这篇文章中，我们继续我们的努力做一个系统的发展的解决理论的共形公式的爱因斯坦约束方程的紧流形上的边界。通过以自然的方式构建我们最近在Holst和Tsogtgerel（2013）以及Holst，Nagy和Tsogtgerel的工作，（2008年，2009年），也对麦克斯韦的工作（2004，2005，2009）和Dain（2004），在合理的假设下，我们对数据证明存在的近和远-常平均曲率解的一类罗宾边界条件通常用于在文献中模拟黑洞，与第三个存在性结果的常平均曲率（CMC）出现作为一个特殊情况。Dain和Maxwell讨论了演化为包含黑洞的时空的初始数据工程，确定了CMC设置中渐近欧几里得流形上的共形公式的解，内部边界条件表示切除的内部黑洞区域。Holst和Tsogtgerel汇编了Dain和Maxwell的内边界结果，然后发展了一般的内条件来模拟Dain和Maxwell的带边界紧致流形的视视界边界条件，并随后证明了具有这种边界条件的紧致流形上Lichnerowicz方程解的存在性。本文在Holst和Tsogtgerel的基础上，讨论了一般的带边界的紧致流形的非CMC情形。与我们以前的文章一样，我们在这里再次关注低正则性数据和不同类型边界条件之间的相互作用。虽然我们在这里的工作主要是为了扩展的解决方案理论的紧凑的边界情况下，我们也开发了几个技术工具，有可能与渐近欧几里德的情况下使用。
英文摘要：
In this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the Einstein constraint equations on compact manifolds with boundary. By building in a natural way on our recent work in Holst and Tsogtgerel (2013), and Holst, Nagy, and Tsogtgerel (2008, 2009), and also on the work of Maxwell (2004, 2005, 2009) and Dain (2004), under reasonable assumptions on the data we prove existence of both near- and far-from-constant mean curvature solutions for a class of Robin boundary conditions commonly used in the literature for modeling black holes, with a third existence result for constant mean curvature (CMC) appearing as a special case. Dain and Maxwell addressed initial data engineering for space-times that evolve to contain black holes, determining solutions to the conformal formulation on an asymptotically Euclidean manifold in the CMC setting, with interior boundary conditions representing excised interior black hole regions. Holst and Tsogtgerel compiled the interior boundary results covered by Dain and Maxwell, and then developed general interior conditions to model the apparent horizon boundary conditions of Dain and Maxwell for compact manifolds with boundary, and subsequently proved existence of solutions to the Lichnerowicz equation on compact manifolds with such boundary conditions. This paper picks up where Holst and Tsogtgerel left off, addressing the general non-CMC case for compact manifolds with boundary. As in our previous articles, our focus here is again on low regularity data and on the interaction between different types of boundary conditions. While our work here serves primarily to extend the solution theory for the compact with boundary case, we also develop several technical tools that have potential for use with the asymptotically Euclidean case.
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