Investigation on Differential Geometry and General Relativity

微分几何与广义相对论研究

• 批准号: 1704393
• 负责人: Xin Zhou
• 金额: $ 5.2万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1704393&HistoricalAwards=false
• 关键词: Investigation; Differential; Geometry; General; Relativity

AbstractAward: DMS 1406337, Principal Investigator: Xin Zhou The physical description of a soap film spanning a wire boundary leads to an important class of surfaces in geometry, called "minimal surface", because the energy-minimization condition that the soap film satisfies corresponds to a surface of least area, at least among nearby surfaces. Also as intrinsic geometric objects of the underlying space, minimal surfaces can be viewed as non-linear analog of the eigenstates in Quantum Mechanics. The mathematical problem that amounts to showing that every bounding wire is spanned by at least one soap film has been studied in detail for many years and admits many important generalizations. Most of the previous techniques for proving such existence properties focused on the "minimizing theory", which corresponds to finding the ground state in Quantum Mechanics. One natural but very difficult technique, called the "min-max theory", corresponding to finding the excited states in Quantum Mechanics (i.e. the eigenstates with energy higher than the ground state), has had striking recent successes, and will be one of the major objects of study in this research program. Another project will study geometric inequalities in general relativity, particularly aiming to understand the mass angular momentum inequalities for axisymmetric initial data sets which model rotating galaxies in astrophysics.More specifically, the principal investigator will study the min-max theory for minimal surfaces via both the geometric measure theory approach and the harmonic map approach. Concerning the geometric measure theory approach, some of the aspects of the min-max theory to be studied in this research program include extensions of the PI's index bound theorem beyond positive Ricci curvature conditions, as well as versions of the Almgren-Pitts min-max theory for more general ambient spaces, including manifolds with boundary and smooth metric measure spaces (where the Bakry-Emery Ricci tensor plays a role). Certain geometric problems related to the Min-max theory will be studied, such as Hersch-type estimates for higher-dimensional minimal hypersurfaces with bounded Morse index. Concerning the min-max theory via harmonic maps, the PI intends to study the issue of Morse index and branch points of the min-max minimal surfaces constructed by Colding-Minicozzi and himself, and the corresponding free boundary problem.

摘要奖：DMS 1406337，首席研究员：周欣：跨越导线边界的肥皂膜的物理描述导致了几何中一类重要的曲面，称为“极小曲面”，因为肥皂膜满足的能量最小化条件对应于一个面积最小的曲面，至少在附近的曲面中是这样。极小曲面也是底层空间的固有几何对象，可以看作是量子力学中本征态的非线性模拟。这个数学问题已经被详细研究了许多年，并承认了许多重要的概括。这个数学问题相当于证明每一条界限线都被至少一层肥皂膜跨越。以往证明这种存在性质的技术大多集中在“极小化理论”上，该理论对应于量子力学中的基态发现。与寻找量子力学中的激发态(即能量高于基态的本征态)相对应的一种自然但非常困难的技术，称为“最小-最大理论”，最近取得了惊人的成功，将成为本研究计划的主要研究对象之一。另一个项目将研究广义相对论中的几何不等式，特别是旨在了解在天体物理中模拟旋转星系的轴对称初始数据集的质量角动量不等式。更具体地说，主要研究人员将通过几何测度论方法和调和映射法研究极小表面的最小-最大理论。关于几何测度论的方法，本研究计划要研究的最小-最大值理论的一些方面包括PI的指标界定理在正Ricci曲率条件之外的扩展，以及更一般环境空间的Almgren-Pitts最小-最大值理论的版本，包括具有边界和光滑度量空间的流形(其中Bakry-Emery Ricci张量起作用)。我们将研究与Min-Max理论有关的某些几何问题，例如具有有界Morse指数的高维极小超曲面的Hersch型估计。关于调和映射的极小极大理论，PI打算研究Colding-Minicozzi和他自己构造的极小极小曲面的Morse指数和分支点问题，以及相应的自由边界问题。