Studies on Einstein Metrics and Related Topics

项目负责人：Michael T. AndersonAnderson和LeBrun都计划研究低维流形的全局黎曼几何，并着眼于将这些结果应用于光滑拓扑、代数几何和理论物理。Anderson将继续研究thurston的几何化猜想，并研究该猜想与广义相对论中真空时空的全局存在性和奇点形成的关系。LeBrun将研究全标量曲率泛函和光滑拓扑的性质之间的关系，以及二次曲率泛函的相关问题。此外，Anderson和LeBrun都计划研究渐近双曲流形的几何主题，以及与弦理论和AdS/ cft对应的当前问题的关系。安德森的研究与黑洞物理学、宇宙的形状和长期演化等问题直接相关。LeBrun的工作研究了霍金量子引力计划引起的基本问题，并将高能物理的Seiberg-Witten方程与引力现象的研究联系起来。

AbstractAward: DMS-0072591Principal Investigator: Michael T. AndersonAnderson and LeBrun both plan to study the global Riemanniangeometry of low-dimensional manifolds, with an eye towards theramifications of these results for smooth topology, algebraicgeometry, and theoretical physics. Anderson will continue work onThurston's geometrization conjecture, and study relations of thisconjecture with global existence and singularity formation forvacuum space-times in general relativity. LeBrun will studyrelations between properties of the total scalar curvaturefunctional and smooth topology, as well as related problemsconcerning quadratic curvature functionals. In addition,Anderson and LeBrun both plan to investigate topics in thegeometry of asymptotically hyperbolic manifolds, and relationswith current issues in string theory and the AdS/CFTcorrespondence.Anderson's investigations are directly related to questionsregarding the physics of black holes and the shape and long-termevolution of the universe. LeBrun's work studies fundamentalquestions arising from Hawking's quantum gravity program, andrelates the Seiberg-Witten equations of high-energy physics tothe study of gravitational phenomena.