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

该项目的目标是研究广义相对论中几个重要的相关猜想。这种引力的几何理论是我们理解宇宙大尺度结构的基础，并且具有许多实际应用，例如全球定位系统（GPS）技术的微调。 PI 将寻求建立与质量、电荷、角动量和地平线面积相关的几何不等式族，以探索宏大的弱宇宙审查猜想。这个猜想断言，每当时空中出现奇点（这是一种普遍现象）时，它们必定总是被笼罩在黑洞事件视界内；这与广义相对论是否是一个适当的决定论理论密切相关。具有对称性的特殊黑洞解（称为静止轴对称和电真空）在我们对该理论的理解中发挥着重要作用，该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 旨在证明这种在五个时空维度上具有奇异（透镜空间）拓扑的黑洞的存在性和唯一性。此外，还将研究引力塌缩和黑洞形成的新标准，即由角动量和/或电荷集中引起的标准。 PI 还将检查拟局部质量的拟议定义，以确定它们在数学和物理上是否可行。基于 Bray 的早期工作，PI 最近完成了一种系统方法，通过将每个彭罗斯型不等式简化为椭圆偏微分方程 (PDE) 的规范系统来处理整个彭罗斯型不等式。因此，这些几何不等式的整个范围都是可以实现的。与这些不等式的某个子类的研究相关的自然副产品涉及构造具有二维双曲空间目标的奇异调和图的一般程序，该奇异调和图自然地产生于四维静止轴对称真空爱因斯坦方程。 G. Weinstein 发起了对具有规定奇点的调和图的研究，我们已经开始开发必要的工具，以充分概括 4 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 已经建立了可以被视为 Bartnik 最小质量延伸猜想的第一步，而我们的方法表明了剩余部分应该是成功的方法。俘获面/环猜想涉及黑洞可能形成的条件，受到高度追捧，但尚未得到充分理解。然而，PI 对彭罗斯型不等式的研究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.