Black Holes, Geometric Inequalities, and Partial Differential Equations

这个项目的目标是研究广义相对论中几个重要的相关理论。爱因斯坦提出的引力几何理论是我们理解宇宙大尺度结构的基础，并有许多实际应用，如全球定位系统（GPS）技术的微调。PI将寻求建立与质量、电荷、角动量和视界面积相关的几何不等式家族，以探索大弱宇宙监督猜想。这个猜想断言，无论何时时空中出现奇点（这是一种普遍现象），它们必须总是被黑洞事件视界所掩盖;这与广义相对论是否是一个正确的决定论密切相关。具有对称性的特殊黑洞解（称为静态轴对称和电真空）在我们理解理论中起着重要作用，本项目试图将它们分类到与弦理论相关的更高维度。此外，还将研究引力坍缩和黑洞形成的新标准，PI还将研究拟局域质量的拟议定义，以确定它们在数学和物理上是否可行。最近，PI与Bray、Kazaras和Stern合作，发现了正质量定理的一个新的简单证明，这一结果自40年前由Schoen、Yau和维滕首次证明以来，在数学相对论中发挥了开创性的作用。这种新的方法，这也被推广到时空和双曲设置，产生一个明确的下界的质量与（时空）调和函数相关的数量，并建议一个策略，建立该定理的约束稳定性或几乎刚性。基于与Bray的初步合作，PI已经完成了一个系统的方法来处理整个Penrose型不等式家族，通过将每个不等式减少到椭圆PDE的规范系统，从而将这些几何不等式的整个范围都放在可及范围内。与山田和温斯坦一起，他们在20世纪90年代开始研究与轴对称4D爱因斯坦方程相关的具有规定奇点的调和映射，PI开发了必要的工具来基本推广4D结果，以允许更高维度的奇异拓扑结构以及广泛的对称空间目标。这项工作表明，它是可能的，以获得完整的分类静态轴对称黑洞在真空和超重力。在与Alaee和Yau的联合工作中，PI已经开始研究基于Wang-Yau准局域质量的建议的贝肯斯坦边界;这些不等式涉及相对论性物体中包含的熵/信息，并具有从热力学到计算机科学的广泛影响。这个初步的研究提供了解决完整的贝肯斯坦猜想的基础，并且通过处理黑洞可能形成的条件的捕获表面/环猜想的方法密切相关。此外，与安德森的联合工作已经建立了可以被认为是Bartnik的最小质量扩展猜想的第一步，PI的方法表明了剩余部分应该是一个成功的策略。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The goal of this project is to study several important related conjectures in general relativity. This geometric theory of gravity proposed by Einstein 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. Furthermore, new criteria for gravitational collapse and black hole formation will be studied, and the PI will also examine proposed definitions of quasi-local mass in order to determine whether they are mathematically and physically viable. Lastly, fundamental questions concerning the shape (topology) of the cosmos will be addressed, including whether we live in a finite or infinite universe.Recently the PI, in collaboration with Bray, Kazaras, and Stern has found a new and simple proof of the positive mass theorem, a result which has played a seminal role in mathematical relativity since its initial proof by Schoen, Yau, and Witten 40 years ago. This new approach, which has also been generalized to the spacetime and hyperbolic settings, yields an explicit lower bound for the mass in terms of quantities associated with (spacetime) harmonic functions, and suggests a strategy to establish the conjectured stability or almost rigidity for this theorem. Based on an initial collaboration with Bray, the PI has completed a systematic approach to treating the full family of Penrose-type inequalities by reducing each to a canonical system of elliptic PDE, thus placing the entire range of these geometric inequalities within reach. Together with Yamada and Weinstein, who initiated the study of harmonic maps with prescribed singularities associated with the axisymmetric 4D Einstein equations in the 1990s, the PI has developed the tools necessary to substantially generalize the 4D results to allow for exotic topologies in higher dimensions as well as a wide range of symmetric space targets. This work suggests that it is possible to obtain the full classification of stationary axisymmetric black holes within vacuum and supergravity. In joint work with Alaee and Yau, the PI has begun a study of the proposed Bekenstein bounds based on the Wang-Yau quasi-local mass; these inequalities concern the entropy/information contained within a relativistic body and have wide ranging implications from thermodynamics to computer science. This initial study has provided the foundations to address the full Bekenstein conjecture, and is closely related via the approach to the trapped surface/hoop conjecture dealing with the conditions under which black holes may form. In addition, joint work with Anderson has established what may be considered as the first step of Bartnik's minimal mass extension conjecture, and the PI's methods indicate what should be a successful strategy for the remaining parts.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.