Canonical metrics on Kahler and Riemannian manifolds and their moduli

构造各种几何结构的适当空间是代数几何和微分几何中的一个基本问题。这些空间被称为模空间，在从分析、几何、拓扑学到数论等数学学科中发挥了重要作用。一个多世纪以前，黎曼、庞加莱和希尔伯特率先研究了曲面上的“规范”度量结构。通过与爱因斯坦方程的联系，正在研究的项目将有助于理解天文学和宇宙学。本项目中分析爱因斯坦型方程的方法也将在工程和经济学中得到应用。首席研究员组织并参与综合研究/教育项目，旨在吸引弱势群体的学生学习更高级的数学学科，如几何分析，从而帮助提高国家的整体教育水平。PI将调查以下项目，所有这些项目都来自模空间的代数/分析方面的研究。首先，他将通过代数和解析方法研究极化Kahler流形的模空间几何（例如，在模空间上线束和高度的正性）。其次，他将研究希尔伯特和芒福德在PI及其合作者的工作基础上建立代数模的几何不变理论（GIT）的有效范围。特别是，为了开发一个比经典GIT更灵活的通用框架，PI研究了更多的示例。第三，他将研究低维流形上Kahler-Ricci流的极限行为，特别是复杂结构跳跃的情况。第四，他将研究黎曼流形在无限维单位球中的正则嵌入，并将嵌入的外在几何与底层黎曼流形上的正则度量联系起来。这是试图统一共形几何和Kahler几何之间的相似性的第一步。上述项目将应用来自代数/微分几何的解析和代数工具，以及首席研究员和他的合作者多年来引入的新思想。

It is a fundamental problem to construct appropriate spaces of various geometric structures in both algebraic and differential geometry. These spaces, referred to as moduli spaces, have played a fundamental role in mathematical subjects ranging from analysis, geometry, and topology to number theory. More than a century ago, Riemann, Poincare, and Hilbert pioneered the study of "canonical" metric structures on a surface. Through the connection to the Einstein equations, the projects under investigation will help understand astronomy and Cosmology. The methods of analyzing Einstein type equations in this project will also lead to applications in Engineering and economics. The principal investigator organizes and participates in the integrated research/education programs that aim to attract students from under-represented groups to the study of more advanced mathematics subjects such as geometric analysis, thus helping to improve the overall education level of the nation. The PI will investigate the following projects, all of which emerge from the study of the algebraic/analytic aspects of the moduli spaces. First, he will study the geometry of the moduli space of polarized Kahler manifolds via algebraic and analytic means (e.g., the positivity of the line bundles and heights over the moduli space). Second, he will study the effective scope of geometric invariant theory (GIT) invented by Hilbert and Mumford on the construction of algebraic moduli based on the work of the PI and his collaborators. In particular, the PI investigates more examples in order to develop a general framework that is more flexible than the classical GIT. Third, he will study the limiting behavior of Kahler-Ricci flow on low-dimensional manifolds, in particular the situation when the complex structure jumps. Fourth, he will study the canonical embedding of Riemannian manifolds into the infinite-dimensional unit sphere and relate the extrinsic geometry of the embedding to the canonical metrics on the underlying Riemannian manifold. This is intended as the first step in attempting to unify the similarities between conformal and Kahler geometry. The projects proposed above will apply both analytic and algebraic tools coming from algebraic/differential geometry as well as the new ideas introduced by the principal investigator and his collaborators over the years.