On the Geometry of Kahler-Einstein Manifolds

University Of California-Irvine, Irvine CA

Investigators

Abstract

ABSTRACT DMS - 0204667. PI: Zhiqin Lu On the Geometry of Kaehler-Einstein Manifold In this project, the proposer is going to understand Kaehler-Einstein metrics, especially Calabi-Yau manifolds using the theory of moduli spaces, through three different ways. The first way is to study the potential function of the Bergman metric of a polarized compact Kaehler manifold. For the application in the Kaehler-Einstein geometry, one needs to estimate the potential function from below for a family of manifolds. The problem is related to the stability of manifolds in the sense of Geometric Invariant Theory. The second way is to study the relation between the K stability and the Mumford stability. Such a relation, if exists, would give one new insights of the relations between the Kaehler-Einstein geometry and the stability of manifolds. The third way is the geometry of moduli space of polarized Calabi-Yau manifolds. A Kaehler metric on the moduli space has been defined and was found that the Ricci curvature of such a metric is negative away from zero by the proposer. In order to introduce geometric analysis to the place, one needs to prove a version of the maximal principle on the moduli space, even if the moduli space, in general, is not smooth. Einstein's general theory of relativity is a theory that interprets the concept of gravity into the geometric property of the space. Recent development of in physics shows that the universe may be of dimension ten, with three dimension in space and one dimension in time plus a tiny six dimensional space called Calabi-Yau threefold. One of the main mathematical tool to study the geometry of the space is differential geometry. Since the discovery of the general relativity, differential geometry becomes crucial to both mathematicians and physicists. The project is one of the main field in differential geometry. It will help a lot in understanding one of the basic force of the universe: the gravity and ultimately understanding the space we are living with. It is difficult to believe that without an extensive study of fundamental sciences such as general relativity, modern technology like the use of atomic energy can come true. By the same reason, today's fundamental study will not only enlarge our knowledge but eventually benefit people's life as well.