Conformal Geometry, Partial Differential Equations, and Mathematical Relativity

共形几何、偏微分方程和数学相对论

• 批准号: 1608782
• 负责人: Jie Qing
• 金额: $ 24.9万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-15 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608782&HistoricalAwards=false
• 关键词: Conformal; Geometry; Partial; Differential; Equations

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.

该项目研究共形几何，即空间上的保角变换集的研究，特别是结合反德西特类型的广义相对论和理论物理学中的共形场论之间的对应关系。这是微分几何和数学物理学科中非常重要和基础的领域。正是由于这种与理论物理学的对应关系，共形几何取得了实质性进展。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。Fefferman 和 Graham 在 20 世纪 80 年代的开创性论文开发了一种环境空间方法来研究共形几何的局部标量不变量，并赋予偏微分方程方法研究共形几何的能力。最近，人们对这种构造的兴趣重新高涨，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之后。为了本着费弗曼和格雷厄姆的精神为这种对应关系奠定数学基础，需要研究共形紧致爱因斯坦流形以及这种对应关系的各种数学解释。对于给定的共形流形，当规定的共形无穷大时，关于是否存在共形紧爱因斯坦流形的最基本问题仍然在很大程度上悬而未决。该项目将研究共形紧致爱因斯坦流形的一般紧致性性质，这将导致更普遍的存在理论。此外，该项目将本着费弗曼和格雷厄姆的精神，开发一种环境空间方法来研究子流形的共形几何。