The Curvature of 4-Manifolds

项目编号：DMS-1205953首席研究员：Claude R. lebrun首席研究员计划研究四维黎曼几何中一系列相关的全局问题。提出的研究的主要目标是发现和发展黎曼4流形的曲率和底层空间的微分拓扑之间的基本联系。研究的主要问题集中在几类规范度量的存在性、唯一性和详细结构上。研究主题将包括爱因斯坦度量，极值Kaehler度量，巴赫平坦度量和自对偶度量。一些相关的问题，在更高的维度也将被考虑，预期发现进一步的基本差异，在四维和更高的维度设置。本研究项目旨在探索微分几何数学领域的基本问题，并发现数学与理论物理之间的新联系。许多计划中的研究活动的灵感来自于目前试图弥合爱因斯坦引力理论与量子场论之间鸿沟的尝试。量子场论描述了微观尺度上的自然力。研究的一个主要焦点是四维宇宙的大尺度（拓扑）结构对爱因斯坦引力场方程的黎曼签名解是否存在的影响。

AbstractAward: DMS-1205953 Principal Investigator: Claude R. LeBrunThe principal investigator plans to study a family of related global problems in 4-dimensional Riemannian geometry. The primary goal of the proposed research is to discover and develop fundamental links between the curvature of a Riemannian 4-manifold and the differential topology of the underlying space. The main problems to be investigated focus on the existence, uniqueness, and detailed structure of several classes of canonical metrics. Topics of study will include Einstein metrics, extremal Kaehler metrics, Bach-flat metrics, and self-dual metrics. Some related problems in higher dimensions will also be considered, in anticipation of the discovery of further fundamental differences between the 4-dimensional and higher-dimensional settings.This research program aims both to explore fundamental issues in the mathematical field of differential geometry, and to discover new links between mathematics and theoretical physics. Much of the planned research activity takes its inspiration from current attempts to bridge the gulf separating Einstein's theory of gravitation from the quantum field theories that describe the forces of nature on a microscopic scale. A major focus of the research is the influence of the large-scale (topological) structure of a 4-dimensional universe on whether or not it admits Riemannian-signature solutions of Einstein's gravitational field equations.