Investigation on Conformally Compact Einstein Manifolds and Related Problems

共形紧爱因斯坦流形及相关问题的研究

• 批准号: 0202122
• 负责人: Gang Tian
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202122&HistoricalAwards=false
• 关键词: Investigation; Conformally; Compact; Einstein; Manifolds

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.

摘要 DMS - 0202122。该项目的目标是研究共形紧致爱因斯坦流形的几何形状和其他相关问题。这些流形是大约十年前由数学家首次研究的。几年前，当人们发现它们是弦理论中新提案 ADS/CFT 对应的数学框架时，出现了新的想法和刺激。因此，共形紧致爱因斯坦流形的研究对于物理学来说甚至变得很重要。作者已经对此类流形的几何进行了研究，但仍有许多问题有待研究。未来作者希望能够解决存在性问题。如果共形无穷大有足够的对称性，我们希望能找到明确的解。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，环境流形的全局几何形状与边界的共形几何形状之间存在着深刻的关系。作者打算进一步探索这个方向。本提案研究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层对应的框架。作者将研究这些流形的各个几何方面以及物理学中出现的问题。对这种特殊类型的非紧流形的研究，其无限远处的几何形状得到很好的控制，也将为研究更一般的非紧流形提供见解和想法。