Geometric Boundary Value Problems in General Relativity

许多自然现象，如液体的运动，固体的弯曲，或温度的传播，可以用偏微分方程（PDE）来描述。例如，电子设备，包括手机或电脑，在正常操作期间产生热量，并且有必要将这些热量传导出去以防止过热。利用具有特定边界条件（如表面温度）的拉普拉斯方程分析温度分布的稳态，确保有效的热传导并防止过热。有趣的是，对我们宇宙结构的研究，受爱因斯坦广义相对论的支配，也产生了类似于拉普拉斯方程的几何偏微分方程。该研究项目旨在研究那些由于量化宇宙有界区域内的质量或能量而产生的偏微分方程，例如冰川系统或二元黑洞。我们的目标是通过揭示几何边值问题和拉普拉斯方程已知性质之间的隐藏联系来推进我们对宇宙结构的理解。该项目还将涉及指导学生和开展教育活动，以提高更广泛受众对STEM的认识。该研究项目将解决长期以来与广义相对论中Bartnik的准局部质量和具有指定边界数据的爱因斯坦流形的存在有关的问题。1989年，Bartnik通过极小化容许扩张中渐近定义的质量，提出了拟局部质量的概念，并给出了几个证明。这些理论与正质量定理、彭罗斯不等式和纯量曲率有着惊人的联系。一种新的方法已被应用到推进静态扩展猜想，并有望加深我们的理解标量曲率的背景下，边界几何和静态真空流形。最近在高维中的定态猜想和pp波反例方面取得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Penrose不等式。此外，这项研究将解决其他几何问题，特别是关于爱因斯坦流形与规定的共形边界度量和平均曲率的存在。这一奖项反映了NSF的法定使命，并已被认为是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的支持。

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.