On the Geometry of Kahler-Einstein Manifolds

摘要DMS - 0204667。关于Kaehler-Einstein流形的几何在这个项目中，提出者将利用模空间理论，通过三种不同的方式来理解Kaehler-Einstein度量，特别是Calabi-Yau流形。第一种方法是研究极化紧化Kaehlermanifold的Bergman度规的势函数。为了在kaehler - einstein几何中的应用，我们需要从下面估计流形族的势函数。这个问题与几何不变理论意义上流形的稳定性有关。第二种方法是研究K稳定性与Mumford稳定性之间的关系。这样的关系，如果存在，将会给Kaehler-Einstein几何和流形稳定性之间的关系提供一个新的见解。第三种方法是极化Calabi-Yau流形的模空间几何。本文定义了模空间上的Kaehlermetric，并证明了该度量的Ricci曲率在远离零的地方是负的。为了在这个地方引入几何分析，我们需要在模空间上证明极大原理的一个版本，即使模空间通常不是光滑的。爱因斯坦的广义相对论是一种将引力概念解释为空间几何特性的理论。物理学的最新发展表明，宇宙可能是十维的，空间是三维的，时间是一维的，加上一个很小的六维空间，称为卡拉比-丘三维空间。研究空间几何的主要数学工具之一是微分几何。自从广义相对论的发现以来，微分几何对数学家和物理学家来说都变得至关重要。该课题是微分几何的主要研究领域之一。它将帮助人们理解宇宙的基本力量之一：引力，并最终理解我们生活的空间。很难相信，没有对广义相对论等基础科学的广泛研究，像原子能这样的现代技术就能成为现实。同样的道理，今天的基础研究不仅会扩大我们的知识，最终也会有益于人们的生活。

ABSTRACT DMS - 0204667.PI: Zhiqin Lu On the Geometry of Kaehler-Einstein ManifoldIn this project, the proposer is going to understandKaehler-Einstein metrics, especially Calabi-Yau manifoldsusing the theory of moduli spaces, through three differentways. The first way is to study the potential function ofthe Bergman metric of a polarized compact Kaehlermanifold. For the application in the Kaehler-Einsteingeometry, one needs to estimate the potential function frombelow for a family of manifolds. The problem is related tothe stability of manifolds in the sense of GeometricInvariant Theory. The second way is to study the relationbetween the K stability and the Mumford stability. Such arelation, if exists, would give one new insights of therelations between the Kaehler-Einstein geometry and thestability of manifolds. The third way is the geometry ofmoduli space of polarized Calabi-Yau manifolds. A Kaehlermetric on the moduli space has been defined and was foundthat the Ricci curvature of such a metric is negative awayfrom zero by the proposer. In order to introduce geometricanalysis to the place, one needs to prove a version of themaximal principle on the moduli space, even if the modulispace, in general, is not smooth.Einstein's general theory of relativity is a theory thatinterprets the concept of gravity into the geometricproperty of the space. Recent development of in physicsshows that the universe may be of dimension ten, with threedimension in space and one dimension in time plus a tinysix dimensional space called Calabi-Yau threefold. One ofthe main mathematical tool to study the geometry of thespace is differential geometry. Since the discovery of thegeneral relativity, differential geometry becomes crucial toboth mathematicians and physicists. The project is one ofthe main field in differential geometry. It will help a lotin understanding one of the basic force of the universe: thegravity and ultimately understanding the space we are livingwith. It is difficult to believe that without an extensivestudy of fundamental sciences such as general relativity,modern technology like the use of atomic energy can cometrue. By the same reason, today's fundamental study will notonly enlarge our knowledge but eventually benefit people'slife as well.