Geometric Inequalities and Partial Differential Equations in General Relativity

这一提议的主要研究主题包括数学相对论中的下列重要猜想：静态/静态度规扩张猜想、完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Bartnik对准局部质量的一个特别有希望的定义，该定义寻求将总质量(或ADM)局域化。虽然这一定义满足了最理想的性质，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造成了严重的限制。为了纠正这个问题，静态/定常度规延拓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/定常真空爱因斯坦方程解的ADM质量。在最近Pi和M.Anderson的共同工作中，在合理的假设下建立了这个边值问题的存在性；这一发展是朝着完全猜想迈出的重要一步。彭罗斯不等式将时空的总质量与其事件视界(黑洞边界)的面积联系在一起：总质量的平方大于或等于事件视界的总面积除以16pi。这可以被看作是对时空的任意类空片的猜想，在这种情况下，一个重要的特殊情况(当片的第二基本形式消失时)已经被Huisken和Ilmanen(一个黑洞)和Bray(有限多个黑洞)独立地证实。在最近与Bray的合作中，PI成功地将这一问题归结为求解一个规范的偏微分方程组，并在特殊情况下证实了该问题的存在性。PI与韩启元一起，全面地分析了该系统的一半的爆破和正则性。这个项目的一个主要目标是通过证明一个普遍存在的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，以便产生一种系统地处理其他相关几何不等式的方法。尽管广义相对论中对准局部质量有许多建议的定义，但大多数似乎都受到这样或那样的困扰。就巴特尼克的质量而言，唯一的问题是其定义的抽象性质。如果静态扩张猜想得到验证，这个单独的困难就应该得到解决，从而为几个应用程序打开了道路。例如，人们普遍认为，当足够多的质量集中在一个太小的区域时，引力崩塌必然随之而来，并导致黑洞--这样的说法通常被称为环状猜想。虽然有许多方法可以准确地描述一个区域的大小，但在建立适当的准局部质量概念之前，这个猜想的严格而完整的版本仍然难以捉摸。适当定义的准局部质量概念的另一个可能的应用是解决爱因斯坦方程的长期存在问题。由于这些方程形成了一个双曲系，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适当的引力场(准局部质量的一个组成部分)的能量概念。此外，与静态/定常延拓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，用来研究这一猜想的技术在研究其他几种情况下的适定边值问题时应该被证明是有用的，例如，带边界的流形上的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可能是最重要的公开问题，即宇宙审查猜想(时空奇点是否总是被黑洞包围)，这与广义相对论作为物理理论的准确性有关。启发式地说，彭罗斯不平等本质上是宇宙审查有效的必要条件。因此，如果彭罗斯不平等得到证实，它将大大增强人们对宇宙审查制度有效性的普遍信念。最后，为完全彭罗斯不等式开发的方法有望提供一种新的强有力的工具，将关于一般初始数据集的问题归结为时间对称的情况，因此将在广义相对论的其他广泛问题中得到广泛的应用。

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.