Geometric Inequalities and Partial Differential Equations in General Relativity

本提案的主要研究主题包括数学相对论中的以下重要猜想：静态/静态度量扩展猜想，全彭罗斯不等式和其他几何不等式，以及关于黑洞形成的Hoop猜想。第一个问题源于巴特尼克对准局部质量的一个特别有希望的定义，该定义旨在定位总质量（或ADM）。尽管这个定义满足了大多数期望的性质，因此有可能非常有用，但其抽象和非建设性的性质在理解和适用性方面产生了严重的限制。为了纠正这一问题，静态/静态度量扩展猜想认为准局部质量可以计算为满足一定几何边界条件的静态/静态真空爱因斯坦方程解的ADM质量。在PI和M. Anderson最近的联合工作中，在一个合理的假设下，建立了该边值问题的存在性；这一进展是迈向完整猜想的重要一步。彭罗斯不等式通过不等式将时空的总质量与其视界（黑洞边界）的面积联系起来：总质量的平方大于或等于视界的总面积除以16 π。这可以被看作是对时空中任意一个类空切片的猜想，在这种情况下，一个重要的特殊情况（当切片的第二种基本形式消失时）已经被Huisken和Ilmanen（一个黑洞）以及Bray（有限多个黑洞）独立地证实了。在最近与布雷的合作中，PI已经成功地将这个问题简化为求解一个典型的偏微分方程系统，并在特殊情况下证实了它的存在性。与Q. Han一起，PI对该系统的一半进行了爆破和正则性分析。本项目的主要目标是通过证明一个一般存在性结果来完成这个程序，从而建立彭罗斯猜想。PI还计划扩展这一简化过程，以便产生一种系统的方法来处理其他相关的几何不等式。尽管在广义相对论中有许多关于准局部质量的定义，但大多数似乎都受到这样或那样的困扰。在巴特尼克质量的例子中，唯一的问题是其定义的抽象性。如果静态扩展猜想得到验证，这个单独的困难应该得到解决，从而为几个应用打开了道路。例如，人们普遍认为，只要足够的质量集中在一个太小的区域，引力坍缩必然会随之而来，并导致黑洞的产生——这种说法通常被称为环猜想。虽然有许多方法可以准确地描述一个区域的大小，但在建立准局部质量的适当概念之前，这个猜想的严格和完整的版本仍然是难以捉摸的。准局部质量概念的另一个可能的应用是爱因斯坦方程的长时间存在性问题。由于这些方程形成了一个双曲系统，人们很容易寻找一种基于能量方法的理论，类似于标量波动方程的经典理论。实现这种方法的一个主要障碍是缺乏引力场能量（准局部质量的一个组成部分）的适当概念。此外，与静态/平稳扩展猜想相关的边值问题实际上是Ricci算子的椭圆边值问题。因此，为研究这一猜想而开发的技术在研究其他几种情况下的适定边值问题时应该是有用的，例如，具有边界的流形上的Ricci流。至于彭罗斯不等式，它最初是由彭罗斯提出的，目的是研究当今广义相对论中最重要的开放性问题，即宇宙审查猜想（是否时空奇点总是被黑洞包围），这与广义相对论作为一种物理理论的准确性有关。启发式地说，彭罗斯不等式本质上是宇宙审查制度成立的必要条件。因此，如果彭罗斯不等式得到证实，它将大大增加对宇宙审查有效性的普遍信念。最后，为完整的彭罗斯不等式开发的方法有望提供一种新的强大工具，用于将有关一般初始数据集的问题减少到时间对称的情况下，因此将有许多应用于广义相对论中的其他问题。

The primary topics of investigation for this proposal include the following important conjectures in Mathematical Relativity: the Static/Stationary Metric Extension Conjecture, the full Penrose Inequality and other geometric inequalities, as well as the Hoop Conjecture concerning black hole formation. The first of these arises from a particularly promising definition of quasilocal mass due to Bartnik, which seeks to localize the total (or ADM) mass. Although this definition satisfies most desired properties, and thus has the potential to be very useful, its abstract and nonconstructive nature yield severe limitations on understanding and applicability. In order to rectify this problem the Static/Stationary Metric Extension Conjecture asserts that the quasilocal mass may be calculated as the ADM mass of a solution to the Static/Stationary Vacuum Einstein equations which satisfies a certain geometric boundary condition. In recent joint work of the PI and M. Anderson, existence for this boundary value problem has been established under a reasonable assumptions; this development is a significant step towards the full conjecture. The Penrose Inequality relates the total mass of a spacetime to the area of its event horizons (boundary of black holes) via the inequality: total mass squared is greater than or equal to the total area of the event horizons divided by 16 pi. This may be viewed as a conjecture for an arbitrary spacelike slice of a spacetime, and in this setting an important special case (when the second fundamental form of the slice vanishes) has been confirmed independently by Huisken and Ilmanen (one black hole) and by Bray (finitely many black holes). In recent joint work with Bray, the PI has succeeded in reducing this problem to solving a canonical system of partial differential equations, and has confirmed existence in special cases. With Q. Han the PI has completely analyzed the blow-up and regularity properties for half of this system. It is a major goal of this project to complete this program by proving a general existence result, and thus establishing the Penrose Conjecture. The PI also plans to extend this reduction process in order to yield a systematic way of treating other related geometric inequalities.Although there are numerous proposed definitions for quasilocal mass in General Relativity, most seem to suffer from one affliction or another. In the case of Bartnik's mass, the sole problem is the abstract nature of its definition. If the Static Extension Conjecture were verified, this lone difficulty should be resolved, thus opening the way for several applications. For example, it is generally believed that whenever enough mass is concentrated in too small of a region, gravitational collapse must ensue and result in a black hole - such a statement is often referred to as the Hoop Conjecture. While there are many ways to accurately describe the size of a region, until a proper notion of quasilocal mass is established, a rigorous and complete version of this conjecture will remain elusive. Another possible application of a properly defined notion of quasilocal mass, is to the long time existence problem for the Einstein Equations. As these equations form a hyperbolic system, it is tempting to search for a theory based on an energy method analogous to the classical theory of the scalar wave equation. A major obstacle to realizing such an approach, is the lack of an appropriate notion of energy for the gravitational field (a constituent of quasilocal mass). Moreover the boundary value problem associated with the Static/Stationary Extension Conjecture, is in fact an elliptic boundary value problem for the Ricci operator. Hence the techniques developed to study this conjecture should prove useful when investigating well-posed boundary value problems in several other settings, such as for instance, the Ricci flow on manifolds with boundary. As for the Penrose Inequality, it was originally put forth by Penrose to study what is perhaps the most important open question in General Relativity today, namely the Cosmic Censorship Conjecture (whether spacetime singularities are always enclosed by black holes), which is related to General Relativity's veracity as a physical theory. Heuristically, the Penrose Inequality is essentially a necessary condition for cosmic censorship to hold. Thus if the Penrose Inequality were to be confirmed it would add significantly to the general belief in the validity of cosmic censorship. Lastly, methods developed for the full Penrose Inequality are expected to provide a new powerful tool for reducing questions concerning general initial data sets to the case of time symmetry, and therefore will have numerous applications to a wide range of other problems in General Relativity.