FRG: Collaborative Research: Analysis of the Einstein Constraint Equations

摘要：DMS 1265187, 1262982, 1263431, 1263544PI: Rafe R Mazzeo，斯坦福大学PI: Michael Holst，加州大学圣地亚哥分校PI: Jim Isenberg，俄勒冈大学PI: David Maxell，阿拉斯加大学该项目的目标是了解在多大程度上可以参数化和构建爱因斯坦演化方程的初始数据集。我们计划利用保形方法的最新进展，在非恒定平均曲率（non-CMC）设置下获得新的存在性结果，了解这些方法的局限性，然后开发用于这些相同目标的替代技术，包括度理论，先验估计和粘合方法。Lichnerowicz方程是保形方法的核心，它是一个半线性椭圆方程。由于其非线性指数的混合符号，它是一种尚未完全理解的类型。完整的Lichnerowicz-Choquet-Bruhat-York方程组是一个更困难的耦合系统，它包含了呈现新的解析微妙之处的特征。最终目标是提供初始数据集的完整参数化，特别是在非cmc设置中，不仅在紧凑背景上，而且对于具有渐近欧几里得，双曲或圆柱形末端的流形，所有这些都与物理应用高度相关。在探索新的方法，我们计划使用新的和先进的分析工具，以及越来越精确和灵活的数值模拟技术。在这个项目的过程中取得的技术进步应该有实质性的应用到许多其他的这种一般类型的方程中，这些方程在纯数学和应用数学以及数学物理的其他部分发挥着重要作用。爱因斯坦的引力场理论是一个非常精确的引力物理数学模型，它在预测和模拟天体物理和宇宙尺度上的引力现象方面做得很好。它与所有已知的引力观测和实验是一致的。从基础数学的角度来看，爱因斯坦的理论涉及两种截然不同的方程。引力场动力学的研究涉及到作为非线性时相关演化偏微分方程系统的爱因斯坦方程的分析，而表示引力态的初始数据集的研究涉及到黎曼几何和作为非线性时相关演化偏微分方程系统的爱因斯坦约束方程的研究。在过去的十年里，人们在理解这两个方程方面取得了显著的进展。这个项目的重点是发展对约束方程的更完整的理解。对爱因斯坦方程的研究是数学和物理学之间一个非常重要的联系点，它一方面推动了微分几何和偏微分方程的许多进步，一方面也提供了一个令人信服的、准确的物理世界模型，无论是在天体物理学还是在宇宙学尺度上。这个项目有可能解决这一领域的重大未决问题。

AbstractAward: DMS 1265187, 1262982, 1263431, 1263544PI: Rafe R Mazzeo, Stanford University PI: Michael Holst, University of California - San Diego PI: Jim Isenberg, University of Oregon PI: David Maxell, University of AlaskaThe goal of this project is to understand the extent to which one can parametrize and construct initial data sets for the Einstein evolution equations. We plan to capitalize on the recent progress using the conformal method to obtain new existence results in the nonconstant mean curvature (non-CMC) setting, to understand the limits of these methods, and then to develop alternate techniques toward these same goals, including degree theory, a priori estiimates, and gluing methods. The Lichnerowicz equation, central to the conformal method, is a semilinear elliptic equation. Due to the mixed sign of its nonlinear exponents, it is of a type not yet fully understood. The full Lichnerowicz-Choquet-Bruhat-York set of equations is a more difficult coupled system which incorporates features presenting new analytic subtleties. The ultimate aim is to provide a complete parametrization of initial data sets, particularly in the non-CMC setting, not only on compact backgrounds but also for manifolds with asymptotically Euclidean, hyperbolic or cylindrical ends, all of which are highly relevant for physical applications. In exploring new methods, we plan to use new and advanced analytical tools, as well as increasingly accurate and flexible numerical simulation techniques. Technical advances made in the course of this project should have a substantial application to many other equations of this general type which play important roles in other parts of pure and applied mathematics and mathematical physics. Einstein's gravitational field theory is a remarkably accurate mathematical model of gravitational physics, which does an excellent job of predicting and modeling gravitational phenomena at both the astrophysical and cosmological scales. It is consistent with every known gravitational observation and experiment. From the point of view of underlying mathematics, Einstein's theory involves two very distinct types of equations. The study of dynamics of gravitational fields involves the analysis of the Einstein equations as a nonlinear system of time-dependent evolution partial differential equations (PDE), while the study of initial data sets representing gravitational states involves Riemannian geometry and the study of the Einstein constraint equations as a nonlinear system of time-independent PDE.The last decade has witnessed remarkable progress in under- standing both equations. This project focuses on developing a more complete understanding of the constraint equations. The study of the Einstein equations presents a very important point of contact between mathematics and physics, one which has motivated many advances in differential geometry and PDE on the one side, and which also has provided a compelling and accurate model of the physical world, both on the astrophysical and on the cosmological scales. This project has the potential for settling significant open questions in this area.