Investigation on Conformally Compact Einstein Manifolds and Related Problems

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

ABSTRACT DMS - 0202122. The goal of this project is to study the geometry of conformally compact Einstein manifolds and other related problems. These manifolds were first studied by mathematicians about ten years 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 explicit solutions. The global uniqueness is also a challenging problem and requires new ideas. Many recent results have shown that there is a profound relationship between the global geometry of ambient manifolds and the conformal geometry of the boundary. The author intends to further explore this direction. This proposal studies a class of geometric objects called conformally compact Einstein manifolds. They are not only mathematically interesting, but also important for physics because they serve as the framework for a deep correspondence in string theory. The author will study various geometric aspects of these manifolds as well as problems arising in physics. The study of this special class of noncompact manifolds whose geometry at infinity is well under control will also provide insights and ideas to study more general noncompact manifolds.