Investigation on Differential Geometry and General Relativity

摘要：对跨越线边界的肥皂膜的物理描述引出了几何上一类重要的表面，称为“最小表面”，因为肥皂膜所满足的能量最小化条件对应于一个面积最小的表面，至少在附近的表面中是这样。同样作为底层空间的内在几何对象，最小曲面可以看作是量子力学中特征态的非线性模拟。数学问题表明每条边界线至少被一层肥皂膜所跨越，这一问题已经被详细研究了许多年，并得到了许多重要的推广。以前证明这种存在性的技术大多集中在“最小化理论”上，这相当于在量子力学中寻找基态。一种自然但非常困难的技术，称为“最小-最大理论”，对应于寻找量子力学中的激发态（即能量高于基态的特征态），最近取得了惊人的成功，并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，特别是旨在理解天体物理学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，首席研究员将通过几何测量理论方法和谐波映射方法研究最小曲面的最小-最大理论。关于几何测度理论方法，本研究计划中要研究的最小-最大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Almgren-Pitts最小-最大理论在更一般的环境空间中的版本，包括带边界的流形和光滑度量测度空间（其中Bakry-Emery Ricci张量起作用）。与最小-最大理论相关的某些几何问题将被研究，例如具有有界莫尔斯指数的高维极小超曲面的hersch型估计。对于调和映射的最小-极大理论，PI打算研究Colding-Minicozzi和他自己构造的最小-极大极小曲面的Morse指数和分支点问题，以及相应的自由边界问题。

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.