CAREER: Geometric Problems in General Relativity

数学广义相对论中的许多基本结果都涉及到全局守恒的物理量和我们宇宙的几何结构之间的相互作用。引力理论的这一研究领域非常活跃，需要来自数学各个领域的新思想。所提出的研究对于理解从爱因斯坦方程和守恒量中可以推导出哪些几何性质是重要的。这些研究项目可能有助于解答广义相对论中的一些基本问题，包括时空正质量定理的刚性和质心的几何表征。这项研究将运用微分几何、偏微分方程和几何分析的技术。该项目的教育活动将培养一系列学生进入几何分析和相关领域。主要的教育活动，包括工作研讨会，几何日和暑期研究生工作坊，将为学生准备必要的背景，开始研究几何分析。这些活动还将吸引东北其他大学的学生和教师参与，并将鼓励大学之间的合作。本课题包括两个研究方向，以更好地理解广义相对论中的全局守恒量及其与几何结构的联系。第一个研究方向涉及爱因斯坦约束方程解的模空间。一个长期的项目是充分了解守恒量，如ADM能量和线性动量，如何在模空间上变化。这个项目的灵感来自于研究正质量猜想的刚性和巴特尼克准局部质量的最小质量扩展。知道守恒量在变形下如何变化是至关重要的。此外，PI还将研究具有优势能量条件的约束方程的局部变形定理。第二个项目涉及质心和角动量以及渐近平坦初始数据集的等周性问题。PI一直在研究质心和角动量的几种不同概念，假设宇称条件。对于更一般的初始数据集，继续研究物理量是可取的。她还打算研究稳定的常平均曲率曲面和与几何质心自然相关的等围曲面。

Many fundamental results in mathematical general relativity concern the interplay between the globally conserved physical quantities and the geometric structure of our universe. This area of research in gravitational theory is highly active and requires new ideas from various fields of mathematics. The proposed research is important in understanding what geometric properties can be deduced from the Einstein equations and from the conserved quantities. The research projects may lead to the answers to some fundamental questions in general relativity, including rigidity of the spacetime positive mass theorem and geometric characterization of the center of mass. The research will employ techniques from differential geometry, partial differential equations, and geometric analysis. The project's educational activities will train a range of students into the field of geometric analysis and related areas. The main educational activities, including the Working Seminar, Geometry Day, and Summer Graduate Workshop, will prepare students for necessary backgrounds to start research in geometric analysis. Those activities will also attract students and faculty from other universities in the Northeast to participate and will encourage collaborations across universities.This research project comprises two research directions to better understand the globally conserved quantities in general relativity and their connections to the geometric structure. The first research direction concerns the moduli space of solutions to the Einstein constraint equations. A long term project is to fully understand how the conserved quantities, such as the ADM energy and linear momentum, vary on the moduli space. This project is inspired by studying rigidity of the positive mass conjecture and minimal mass extension of Bartnik's quasi-local mass. It is crucial to know how the conserved quantities vary under deformations. In addition, the PI will also study local deformation theorems for the constraint equations with dominant energy condition. The second project concerns the center of mass and angular momentum and the question of isoperimetry in asymptotically flat initial data sets. The PI has been investigating several different notions of center of mass and angular momentum, assuming parity conditions. It is desirable to continue to investigate the physical quantities for more general initial data sets. She also intends to study the stable constant mean curvature surfaces and isoperimetric surfaces that are naturally related to the geometric center of mass.