Moduli Space of Canonical Metrics on Four-Manifolds

一个多世纪以来,数学家专门从事微分几何的主题工作的问题是最好的几何形状(即形状),一个给定的空间可以有:就像一个皮球可以塑造成一个完美的球体,曲率相同数量的点,四面八方,所有二维空间可以做成高度对称的和统一的几何图形,类似的所有三维空间可以切成块,把这样的几何图形。但是相应的问题对于四维空间是完全开放的。此外，这个维度在其他维度中脱颖而出，因为某些四维空间可以支持无限多不同的光滑结构，这些结构可能允许具有完全不同性质的几何形状。这些事实使得四维几何在许多方面不可预测，成为跨学科研究的肥沃领域，因为该学科在许多方向上都有发展，并分支到数学和物理的其他领域。该研究项目基于几何、拓扑和分析等领域之间的相互作用。研究者将对一类无界四维空间进行深入分析，这些空间从广义相对论的角度来看是很重要的，它们具有以复数为模型的几何形状，其几何形状是均匀的。在另一个方向上，研究者将检查曲率同样均匀的封闭四维空间。更详细地说，研究者将对渐近局部欧几里得Kähler流形进行全面的分析。这是一类特殊的非紧化四维流形，它推广了理论物理中的引力瞬子。该项目在理解闭合常数标量曲率Kähler或almost-Kähler流形的整体结构方面具有深远的应用。第二组项目侧重于研究4流形的黎曼性质，并结合来自Seiberg-Witten理论的技术。特别地，研究者将分析爱因斯坦度量或常标量曲率度量的模空间及其在Gromov-Hausdorff拓扑中的紧化。所采用的技术来自规范理论、全局分析、代数几何和微分拓扑。

For more than a century, mathematicians specializing in the subject of differential geometry have worked on the question of what is the best geometry (i.e., shape) that a given space can have: Just as a beachball can be shaped into a perfect sphere, with the same amount of curvature at all points and in all directions, all 2-dimensional spaces can be shaped into highly symmetric and uniform geometries, and similarly all 3-dimensional spaces can be cut into pieces that carry such geometries. But the corresponding question is completely open for 4-dimensional spaces. Moreover, this dimension stands out among others, since certain 4-dimensional spaces can support infinitely many different smooth structures, which may admit geometries with radically different properties. These facts make 4-dimensional geometry in many ways unpredictable, and a fertile field for interdisciplinary research, as the subject has grown in many directions and branched into other areas of mathematics and physics. This research project is based on interactions between fields including geometry, topology, and analysis. The investigator will undertake an in-depth analysis of a class of unbounded 4-dimensional spaces that are important from the perspective of general relativity, that carry a geometry modeled on the complex numbers, and whose geometry is mildly uniform. In another direction, the investigator will examine closed 4-dimensional spaces whose curvature is similarly uniform. In more detail, the investigator will undertake a comprehensive analysis of the asymptotically locally Euclidean Kähler manifolds. This is a special class of non-compact 4-dimensional manifolds that generalizes the gravitational instantons from theoretical physics. The project has far-reaching applications in understanding the global structure of closed constant scalar curvature Kähler or almost-Kähler manifolds. A second group of projects focuses on the study of the Riemannian properties of 4-manifolds in conjunction with techniques coming from Seiberg-Witten theory. In particular, the investigator will analyze the moduli space of Einstein metrics or constant scalar curvature metrics and their compactifications in Gromov-Hausdorff topology. The techniques employed come from gauge theory, global analysis, algebraic geometry, and differential topology.