Topics in Riemannian and Complex Geometry

主要研究人员：Michael Anderson和Claude LeBrunAnderson和LeBrun将探索低维流形的几何和拓扑之间的新关系。本研究的主题是标量曲率泛函与流形拓扑的关系。在3维，Anderson将继续他的计划，从这个角度证明Thurston的几何化猜想，并研究它与4流形上的爱因斯坦度量之间的关系。在第4维，LeBrun将研究标量曲率泛函、Weyl曲率、Seiberg-Witten理论和爱因斯坦度量的存在性之间的关系。勒布朗还将研究更高维度的相关问题。标量曲率泛函是由爱因斯坦和希尔伯特提出的，在爱因斯坦的广义相对论中起着重要的作用。安德森的研究对黑洞物理学和宇宙形状的问题有直接的影响。以类似的方式，LeBrun的工作研究了由霍金的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。

AbstractProposal: DMS 9802722Principal Investigators: Michael Anderson and Claude LeBrunAnderson and LeBrun will explore new relations between the geometries and topologies of low-dimensional manifolds. The main theme of this research is the relation between the scalar curvature functional and manifold topology. In dimension 3, Anderson will continue his program to prove Thurston's geometrization conjectures from this point of view, and study relations between this and Einstein metrics on 4-manifolds. In dimension 4, LeBrun will study relations between the scalar curvature functional, Weyl curvature, Seiberg-Witten theory, and the existence of Einstein metrics. LeBrun will also work on related problems in higher dimensions.The scalar curvature functional was introduced by Einstein and Hilbert, and plays a fundamental role in Einstein's theory of General Relativity. Anderson's investigations have a direct bearing on questions regarding the physics of black holes and the shape of the universe. In a similar way, LeBrun's work studies fundamental questions arising from Hawking's quantum gravity program, and ties the Seiberg-Witten equations of high energy physics to the study of gravitational phenomena.