Geometric Boundary Value Problems in General Relativity

许多自然现象，如液体的运动、固体的弯曲或温度的扩散，都可以用偏微分方程来描述。例如，包括手机或电脑在内的电子设备在正常运行过程中会产生热量，有必要将这些热量传导出去，以防止过热。采用拉普拉斯方程，在给定表面温度等边界条件下，分析温度分布的稳态，保证热传导效率，防止过热。有趣的是，在爱因斯坦广义相对论的指导下，对宇宙结构的研究也产生了类似于拉普拉斯方程的几何偏微分方程。这个研究项目旨在研究那些在宇宙的有限区域（如冰川系统或双黑洞）内量化质量或能量而产生的偏微分方程。其目标是通过揭示几何边值问题与拉普拉斯方程已知性质之间的隐藏联系，推进我们对宇宙结构的理解。该项目还包括指导学生和开展教育活动，以提高广大受众对STEM的认识。该研究项目将解决与广义相对论中的巴特尼克准局部质量和具有规定边界数据的爱因斯坦流形的存在有关的长期猜想。1989年，Bartnik通过最小化可容许扩展中的渐近定义质量，提出了准局部质量的概念，并提出了几个猜想。这些猜想与正质量定理、彭罗斯不等式和标量曲率有着惊人的联系。一种新的方法被应用于推进静态扩展猜想，并有望加深我们对边界几何和静态真空流形背景下的标量曲率的理解。最近在平稳猜想和高维pp波反例方面的重大进展有望阐明各种质量刚性问题，如双曲正质量定理和一般彭罗斯不等式。此外，本研究将解决其他几何问题，特别是关于爱因斯坦流形的存在规定的共形边界度量和平均曲率。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many natural phenomena, such as the movement of liquids, the bending of solids, or the spread of temperature, can be described by partial differential equations (PDEs). For instance, electronic devices, including cell phones or computers, generate heat during normal operation, and it is necessary to conduct this heat away to prevent overheating. The Laplace equation, with specified boundary conditions like surface temperature, is used to analyze the steady-state of temperature distribution, ensuring efficient heat conduction and preventing overheating. Intriguingly, the study of our universe’s structure, governed by Einstein's general relativity, also gives rise to geometric PDEs similar to the Laplace equation. This research project aims to investigate those PDEs that arise from quantifying the mass or energy within bounded regions of the universe, such as glacial systems or binary black holes. The goal is to advance our understanding about the universe's structure by revealing hidden connections between the geometric boundary value problems and the known properties of the Laplace equation. The project will also involve mentoring students and conducting educational activities to enhance STEM awareness among a broader audience. The research project will address longstanding conjectures related to Bartnik’s quasi-local mass in general relativity and the existence of Einstein manifolds with prescribed boundary data. In 1989, Bartnik proposed a notion of quasi-local mass by minimizing the asymptotically defined masses among admissible extensions and raised several conjectures. Those conjectures have led to surprising connections with the positive mass theorem, the Penrose inequality, and scalar curvature. A novel approach has been applied to advance the Static Extension Conjecture and is expected to deepen our understanding of scalar curvature in the context of boundary geometry and the static vacuum manifolds. The recent significant progress toward the Stationary Conjecture and the pp-wave counter-examples in higher dimensions is anticipated to illuminate various mass rigidity problems, such as the hyperbolic positive mass theorem and general Penrose inequality. In addition, this research will resolve other geometric problems, particularly concerning the existence of Einstein manifolds with prescribed conformal boundary metrics and mean curvature.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.