Investigation on Conformally Compact Einstein Manifolds and Related Problems

摘要DMS - 0202122。本课题的目的是研究共形紧致爱因斯坦流形的几何及其他相关问题。这些流形最初是由数学家在大约十年前研究的。几年前，当人们发现它们是弦理论中ADS/CFT对应的新提议的数学框架时，新的想法和刺激出现了。因此，对共形紧致爱因斯坦流形的研究甚至成为物理学的重要课题。作者对这类流形的几何性质做了一些研究，但仍有许多问题有待研究。在未来，作者希望解决存在的问题。如果正形无穷有足够的对称性，人们希望找到显式解。全球独特性也是一个具有挑战性的问题，需要新的想法。最近的许多结果表明，在环境流形的整体几何与边界的共形几何之间存在着深刻的关系。笔者拟就此方向作进一步探讨。本文研究了一类称为共形紧致爱因斯坦流形的几何对象。它们不仅在数学上很有趣，而且对物理学也很重要，因为它们是弦理论中深度对应的框架。作者将研究这些流形的各种几何方面以及在物理上产生的问题。对这类特殊的非紧流形的研究也将为研究更一般的非紧流形提供见解和思路。

ABSTRACT DMS - 0202122.The goal of this project is to study the geometry of conformallycompact Einstein manifolds and other related problems.These manifolds were first studied by mathematicians about tenyears ago. New ideas and stimuli came up a couple of years ago when it was found that they are the mathematical framework for the new proposal ADS/CFT correspondence in string theory. Therefore the study of conformally compact Einstein manifolds has even become important for physics. The author has done work on the geometry of such manifolds, but there remain many problems to be studied. In the future the author hopes to tackle the problem of existence.If the conformal infinity has enough symmetry one hopes to find explicitsolutions. The global uniqueness is also a challenging problem andrequires new ideas. Many recent results have shown that there is aprofound relationship between the global geometry of ambient manifoldsand the conformal geometry of the boundary. The author intends to further explore this direction.This proposal studies a class of geometric objects called conformallycompact Einstein manifolds. They are not only mathematically interesting,but also important for physics because they serve as the framework fora deep correspondence in string theory. The author will study variousgeometric aspects of these manifolds as well as problems arising inphysics. The study of this special class of noncompact manifolds whosegeometry at infinity is well under control will also provide insightsand ideas to study more general noncompact manifolds.