[论文解读] Non-CMC Solutions of the Einstein Constraint Equations on Compact Manifolds with Apparent Horizon Boundaries

Michael Holst, Caleb Meier+1 more|arXiv (Cornell University)|Oct 8, 2013

Geometric Analysis and Curvature Flows参考文献 20被引用 7

一句话总结

本文在具有边界（特别是通过类罗宾型边界条件建模黑洞内部的似量曲面）的紧致黎曼流形上，建立了非常平均曲率（非CMC）解的存在性。通过Schauder型不动点框架与全局障碍构造，证明了近似CMC与远离CMC情形下的存在性，拓展了以往的CMC结果，并为数值相对论应用奠定了基础解理论。

ABSTRACT

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.

研究动机与目标

• 将爱因斯坦约束方程共形形式的解理论从紧致流形边界上的常平均曲率（CMC）情形推广至非CMC情形。
• 通过在内边界上采用类罗宾型边界条件，将黑洞内部模型化于数值相对论中。
• 在低正则性假设下，为弱解建立严格的解存在性框架，避免限制非线性耦合强度的近似CMC假设。
• 通过建立适用于紧致与渐近平坦情形的工具，弥合闭流形结果与渐近欧几里得设定之间的差距。

提出的方法

• 基于约束映射与函数空间不变性的拓扑性质，采用Schauder型不动点论证。
• 通过构造全局下解与上解，确保哈密顿约束方程中Picard迭代映射的不变集。
• 在度量共形变换下，应用哈密顿约束的共形不变性框架，通过系数变换规则保持边界条件。
• 引入一种改进的障碍构造方法，用于处理任意平均曲率与数据的非恒定障碍，无需近似CMC假设。
• 利用迹与Sobolev型空间（如 $W^{s,p}$）且 $s > n/p$，确保在低正则性数据下的适定性。
• 建立罗宾型边界条件系统在共形重标度下的共形不变性，这对于保持物理一致性至关重要。

实验结果

研究问题

• RQ1 能否证明在具有似量曲面边界条件的紧致流形上，爱因斯坦约束方程存在非CMC解？
• RQ2 如何在不依赖近似CMC假设的前提下构造全局障碍，该假设限制了非线性耦合的强度？
• RQ3 共形不变性在度量重标度下，对保持哈密顿约束与边界条件结构方面起什么作用？
• RQ4 低正则性数据如何影响具有混合边界条件的约束方程的可解性？
• RQ5 为紧致流形开发的解框架能否适用于渐近欧几里得设定？

主要发现

• 本文证明了在具有似量曲面边界的紧致流形上，对近似CMC与远离CMC的平均曲率数据，爱因斯坦约束方程存在弱解。
• 通过构造全局下解与上解，确保Picard映射的不变性，从而可应用Schauder不动点定理。
• 建立了哈密顿约束与类罗宾型边界条件系统下的共形不变性结果，表明该系统在共形重标度下变换一致。
• 通过将非线性耦合约束隔离于全局障碍构造中，该方法避免了近似CMC假设，从而具有更广泛的应用范围。
• 分析表明，即使平均曲率不被其梯度有界，解依然存在，扩展了以往仅限于CMC结果的结论。
• 所开发的技术工具，如流形边界上Sobolev空间的迹与嵌入结果，被证明可推广至渐近欧几里得设定。