Conformal Geometry, Partial Differential Equations, and Mathematical Relativity

本课题研究保角几何，研究空间上的保角变换集合，特别是结合理论物理中反德西特型广义相对论与保角场论之间的对应关系。这是微分几何和数学物理中一个非常重要和基础的领域。在共形几何中，由于这种与理论物理的一致性，已经取得了实质性的进展。这些项目的结果将带来更多数学和物理之间的相互作用，从而更好地理解低维空间的几何结构。Fefferman和Graham在20世纪80年代的开创性论文发展了一种环境空间方法来研究保形几何的局部标量不变量，并赋予了偏微分方程方法来研究保形几何的能力。最近，特别是在反德西特型广义相对论与理论物理学中的共形场论之间的对应关系相互作用之后，对这种结构的新兴趣激增。要按照费弗曼和格雷厄姆的精神为这种对应建立数学基础，就需要研究共形紧致爱因斯坦流形以及对这种对应的各种数学解释。对于给定的共形流形在规定的共形无穷大下是否存在共形紧致爱因斯坦流形，这个最基本的问题仍然悬而未决。该项目将研究共形紧致爱因斯坦流形的一般紧性，这将导致更一般的存在理论。此外，该项目将发展一种环境空间方法来研究子流形的共形几何，以Fefferman和Graham的精神。

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.