The Complex and Conformal Geometry of 4-Manifolds

在广义相对论中，爱因斯坦的方程通过约束四维时空的几何形状来控制引力场。这些方程是一个重要的几何偏微分方程家族的原型，将在“黎曼”设置下进行研究，在黎曼设置中没有空间和时间的区别。资助研究的一个主要目标将是精确地确定哪些四维空间包含这些方程的解。这一领域的进展可能会对纯数学以外的领域产生重大影响，特别是因为许多要研究的几何问题与经典和量子引力以及理论物理的其他领域的问题有着内在的联系。研究生将通过参与研究活动得到训练。更具体地说，该项目侧重于四维全局黎曼几何中的一系列相关问题，使用复几何、共形几何、黎曼几何、凯勒几何和辛几何技术。研究的主题包括爱因斯坦度量，爱因斯坦-麦克斯韦方程，ALE流形，标量平坦Kaehler度量，巴赫平坦4流形，庞加莱-爱因斯坦度量和各种扭扭构造。主要目标是发现和阐明曲率和拓扑之间的基本联系，特别是在四维空间中。一个重点是紧流形上正则度量的构造，主要是通过Kaehler和保形方法。另一个重点是理解解的相关模空间，以及相关的冒泡现象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the general theory of relativity, Einstein's equations govern the gravitational field by constraining the geometry of a 4-dimensional space-time. These equations are the prototype for an important family of geometric partial differential equations that will be investigated under this grant, in the "Riemannian" setting where there is no difference between space and time. A principal objective of the funded research will be to determine precisely which 4-dimensional spaces carry solutions of these equations. Progress in this area could have significant repercussions outside of pure mathematics, especially because many of the geometrical questions to be investigated are intrinsically linked to problems in classical and quantum gravity, as well to other areas of theoretical physics. Graduate students will be trained through participation in the research activities.More specifically, this project focuses on a family of related problems in 4-dimensional global Riemannian geometry, using techniques of complex, conformal, Riemannian, Kaehler, and symplectic geometry. Topics to be investigated include Einstein metrics, the Einstein-Maxwell equations, ALE manifolds, scalar-flat Kaehler metrics, Bach-flat 4-manifolds, Poincare-Einstein metrics, and various twistor constructions. The main goal is to discover and elucidate fundamental links between curvature and topology, especially in dimension four. One focus is the construction of canonical metrics on compact manifolds, primarily by means of Kaehler and conformal methods. Another focus is on understanding the relevant moduli spaces of solutions, along with associated bubbling phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.