The Riemannian Geometry of 4-Manifolds

爱因斯坦的方程起源于广义相对论，在广义相对论中，它们通过限制四维时空的几何形状来控制引力场。由该基金资助的研究将试图加深我们对这些方程的“黎曼”版本的理解，在这种情况下，空间和时间方向没有区别。一个中心问题是理解这些方程的两个解在一个封闭的四维空间上是否总是可以连续地相互变形。另一个关键问题是完全理解哪些四维空间包含这些方程的解。这些问题和相关问题的答案不仅在纯数学领域，而且在理论物理学领域都具有根本性的意义，因为这些问题与量子引力理论有关。首席研究员将研究四维全局黎曼几何中的一系列相关问题，重点研究光滑紧致4流形上爱因斯坦度量和其他巴赫平坦度量的存在性、模量和详细结构。本研究计划的主要目标是发现和阐明黎曼4流形的曲率与底层空间的微分拓扑之间的基本联系。发生在维度4中的几何和拓扑之间的相互作用与发生在其他维度上的相互作用在性质上是不同的，对这种现象的更深入理解可能会对微分拓扑产生重大影响。因此，当在更广泛的背景下构建关键问题变得有帮助时，也将考虑高维几何中的相关问题。

Einstein's equations originated in the general theory of relativity, where they govern the gravitational field by constraining the geometry of 4-dimensional space-time. The research funded by this grant will attempt to deepen our understanding of the "Riemannian" version of these equations, a context in which there is no difference between space and time directions. One central problem is to understand whether two solutions of these equations on a closed 4-dimensional space can always be continuously deformed into one another. Another key problem is to completely understand which 4-dimensional spaces carry solutions of these equations. Answers to these and related questions would be of fundamental interest not only in pure mathematics, but also in theoretical physics, where such problems arise in connection with theories of quantum gravity.The principal investigator will study a family of related problems in 4-dimensional global Riemannian geometry, focusing on existence, moduli, and detailed structure of Einstein metrics and other Bach-flat metrics on smooth compact 4-manifolds. The primary goal of this research program is to discover and elucidate fundamental links between the curvature of a Riemannian 4-manifold and the differential topology of the underlying space. The interplay between geometry and topology that occurs in dimension four is qualitatively different from what occurs other dimensions, and a deeper understanding of this phenomenon could have significant repercussions in differential topology. Related problems in higher-dimensional geometry will therefore also be considered when it becomes helpful to frame key issues in a broader context.