Conformal Geometry, Partial Differential Equations, and Mathematical Relativity

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

The project studies conformal geometry, the study of the set of angle-preserving transformations on a space, particularly in conjunction with the correspondence between general relativity of anti-de Sitter type and conformal field theory in theoretic physics. This is a very important and fundamental area in the subjects of differential geometry and mathematical physics. There has been substantial progress in conformal geometry that is motivated by this correspondence with theoretic physics. As a consequence of these projects, more interactions between mathematics and physics will be brought out, thereby achieving better understanding of geometric structure of low dimensional spaces. Fefferman and Graham's seminal paper in the 1980s developed an ambient space approach to study local scalar invariants for conformal geometry and empowered the partial differential equation approach to the study of conformal geometry. The renewed interests in such a construction have surged recently, particularly after the interaction of the correspondence between general relativity of anti-de Sitter type and conformal field theory in theoretic physics. To develop a mathematical foundation for this correspondence in the spirit of Fefferman and Graham requires the study of conformally compact Einstein manifolds and various mathematical interpretations of such a correspondence. The most fundamental question on the existence of conformally compact Einstein manifold for a given conformal manifold as the prescribed conformal infinity remains largely open. The project will investigate the general compactness property for conformally compact Einstein manifolds, which will lead to more general existence theory. Furthermore, the project will develop an ambient space approach to studying the conformal geometry of submanifolds in the spirit of Fefferman and Graham.