Conformal Geometry, Partial Differential Equations, and Mathematical Relativity

该项目研究共形几何，研究空间上的保角变换集，特别是与理论物理中的反德西特型广义相对论和保角场论之间的对应关系。在微分几何和数学物理中，这是一个非常重要和基本的领域。在与理论物理的这种一致性的推动下，保形几何已经取得了实质性的进展。这些项目的结果是，数学和物理之间将产生更多的相互作用，从而更好地理解低维空间的几何结构。20世纪80年代，Fefferman和Graham的开创性论文发展了一种环境空间方法来研究共形几何的局部标量不变量，并使偏微分方程法能够研究共形几何。最近，人们对这种结构的新兴趣激增，特别是在理论物理中反德西特类型的广义相对论和保形场理论之间的对应关系相互作用之后。为了按照Fefferman和Graham的精神为这种对应建立一个数学基础，需要研究共形紧致爱因斯坦流形和对这种对应的各种数学解释。关于给定的共形流形是否存在共形紧致爱因斯坦流形这一最基本的问题，在很大程度上仍然悬而未决。该项目将研究共形紧的爱因斯坦流形的一般紧性，这将导致更普遍的存在理论。此外，该项目将根据Fefferman和Graham的精神开发一种环境空间方法来研究子流形的共形几何。

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.