Geometric Boundary Value Problems in General Relativity

广义相对论中的几何边值问题

• 批准号: 2304966
• 负责人: Lan-Hsuan Huang
• 金额: $ 35.04万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-12-01 至 2026-11-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304966&HistoricalAwards=false
• 关键词: Geometric; Boundary; Value; Problems; General

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.

许多自然现象，例如液体的运动、固体的弯曲或温度的传播，都可以通过偏微分方程 (PDE) 来描述。例如，包括手机或计算机在内的电子设备在正常运行期间会产生热量。 ，并且有必要将这些热量传导出去以防止过热。有趣的是，该研究使用具有特定边界条件（例如表面温度）的拉普拉斯方程来分析温度分布的稳态，以确保有效的热传导并防止过热。的我们的宇宙结构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因量化宇宙有界区域（例如冰川系统或冰川系统）内的质量或能量而产生的偏微分方程。该项目的目标是通过揭示几何边值问题和拉普拉斯方程的已知属性之间的隐藏联系来增进我们对宇宙结构的理解。该项目还将包括指导学生和开展教育活动以增强理解。该研究项目将解决与广义相对论中的准局域质量以及具有规定边界数据的爱因斯坦流形的存在相关的长期猜想。 1989 年，巴特尼克通过最小化准局域质量提出了准局域质量的概念。在可接受的扩展中渐近地定义了质量，并提出了一些猜想，这些猜想与正质量定理、彭罗斯不等式和一种新颖的方法已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。最近在静态猜想和 pp 波方面取得了重大进展。更高维度的反例有望阐明各种质量刚性问题，例如双曲正质量定理和一般彭罗斯不等式。此外，这项研究还将解决其他几何问题，特别是有关存在性的问题。具有规定的共形边界度量和平均曲率的爱因斯坦流形。该奖项反映了 NSF 的法定使命，并且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，被认为值得支持。