Mass, Geometric Inequalities, and Partial Differential Equations in General Relativity

这个项目的目标是研究广义相对论中几个重要的相关猜想。重力的几何理论是我们理解宇宙大尺度结构的基础，并且有许多实际应用，例如全球定位系统（GPS）技术的微调。PI将寻求建立有关质量、电荷、角动量和视界面积的几何不等式家族，以探索大弱宇宙审查猜想。这个猜想断言，无论何时奇点在时空中出现（这是一种普遍现象），它们一定总是被黑洞事件视界所笼罩；这与广义相对论是否是一个恰当的确定性理论密切相关。具有对称性的特殊黑洞解（称为静止轴对称和电真空）在我们对理论的理解中起着重要作用，本项目试图在与弦理论相关的更高维度上对它们进行分类。特别是，PI旨在证明具有奇异（透镜空间）拓扑的黑洞在五个时空维度上的存在性和唯一性。此外，还将研究引力坍缩和黑洞形成的新准则，即角动量和/或电荷的集中。PI还将研究拟局部质量的定义，以确定它们在数学和物理上是否可行。基于与Bray的早期工作，PI最近完成了一种系统的方法，通过将每个不等式简化为椭圆偏微分方程（PDEs）的正则系统来处理penrose型不等式的整个家族。因此，这些几何不等式的整个范围是可以达到的。一个自然的副产品，与这些不等式的某个子类的研究有关，涉及构造具有二维双曲空间目标的奇异调和映射的一般过程，它自然地产生于四维轴对称真空爱因斯坦方程。温斯坦（G. Weinstein）是研究这种具有规定奇点的调和映射的先驱，我们与他一起，开始开发必要的工具，以大量推广四维结果，以允许更高维度的奇异拓扑以及广泛的对称空间目标。此外，在与M. Anderson的合作中，PI已经建立了可以被认为是巴特尼克最小质量扩展猜想的第一步，我们的方法表明了对其余部分应该是成功的方法。关于黑洞可能形成的条件的俘获面/环猜想受到高度追捧，但尚未得到很好的理解。然而，PI在彭罗斯型不等式上的工作提出了新的黑洞形成标准，以及相对论体的某些等周型不等式。

The goal of this project is to study several important related conjectures in general relativity. This geometric theory of gravity is fundamental to our understanding of the large-scale structure of the universe, and has many practical applications such as to the fine tuning of global positioning system (GPS) technology. The PI will seek to establish families of geometric inequalities relating mass, charge, angular momentum, and horizon area, which probe the grand weak cosmic censorship conjecture. This conjecture asserts that whenever singularities arise in spacetime (which is a generic phenomenon) they must always be shrouded inside a black hole event horizon; this is intimately tied to whether general relativity is a proper deterministic theory. Special black hole solutions with symmetry (referred to as stationary axisymmetric and electro-vacuum) play a large role in our understanding of the theory, and this project seeks to classify them in higher dimensions relevant to string theory. In particular, the PI aims to prove existence and uniqueness for such black holes with exotic (lens space) topologies in five spacetime dimensions. Furthermore, new criteria for gravitational collapse and black hole formation will be studied, namely those due to concentration of angular momentum and/or charge. The PI will also examine proposed definitions of quasi-local mass in order to determine whether they are mathematically and physically viable.Based on earlier work with Bray, the PI has recently completed a systematic approach to treating the full family of Penrose-type inequalities by reducing each to a canonical system of elliptic partial differential equations (PDEs). Thus, the entire range of these geometric inequalities is within reach. A natural by-product, associated with the study of a certain subclass of these inequalities, concerns a general procedure for constructing singular harmonic maps having 2-dimensional hyperbolic space target, which naturally arise from the stationary axisymmetric vacuum Einstein equations in four dimensions. Together with G. Weinstein, who initiated the study of such harmonic maps with prescribed singularities, we have begun development of the tools necessary to substantially generalize the 4-dimensional results to allow for exotic topologies in higher dimensions as well as a wide range of symmetric space targets.  Moreover, in joint work with M. Anderson, the PI has established what may be considered as the first step of Bartnik's minimal mass extension conjecture, and our methods indicate what should be a successful approach to the remaining parts. The trapped surface/hoop conjecture, dealing with the conditions under which black holes may form, is highly sought after but not well understood. However, the PI's work on Penrose-type inequalities suggests new black hole formation criteria as well as certain isoperimetric-type inequalities for relativistic bodies.