On the Geometry of Kahler-Einstein Manifolds

摘要DMS-0204667.PI：陆志勤关于Kaehler-Einstein流形的几何在这个项目中，作者将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致Kaehler流形的Bergman度规的势函数。为了在凯勒-爱因斯坦计量学中的应用，需要估计一族流形的势函数。这个问题与几何不变理论意义下的流形的稳定性有关。第二种方法是研究K稳定性与Mumford稳定性之间的关系。如果存在这样的关系，将会给人们提供一个关于凯勒-爱因斯坦几何和流形稳定性之间关系的新见解。第三种方法是极化Calabi-Yau流形的模空间的几何。定义了模空间上的Kaehler型度量，并证明了这种度量的Ricci曲率不为零。为了将几何分析引入到空间中，人们需要证明模空间上的最大原理的一个版本，即使模空间通常不是光滑的。爱因斯坦的广义相对论是一个将引力概念解释为空间几何性质的理论。物理学的最新发展表明，宇宙可能是十维的，在空间上有三维，在时间上有一维，加上一个微小的六维空间，称为Calabi-Yau三重空间。研究空间几何的主要数学工具之一是微分几何。自从广义相对论被发现以来，微分几何对数学家和物理学家来说都变得至关重要。该项目是微分几何的主要研究领域之一。它将帮助一个人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，如果没有对广义相对论等基础科学的广泛研究，就会出现像使用原子能这样的现代技术。出于同样的原因，今天的基础研究不仅将扩大我们的知识，而且最终也将造福于人们的生活。

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.