Einstein Manifolds and Related Geometric Structures

爱因斯坦流形及相关几何结构

• 批准号: RGPIN-2020-05824
• 负责人: Wang, Mckenzie
• 金额: $ 1.97万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759162
• 关键词: Einstein; Manifolds; Related; Geometric; Structures

A Riemannian metric is a mathematical device that allows us to compute lengths and angles between tangent vectors in n-dimensional spaces.  The Riemann curvature tensor is the basic object derived from a metric for local comparisons of its geometry with that of flat space. There is a part of the Riemann curvature tensor that is of the same mathematical type as the metric g--the Ricci tensor Ric(g). Since conditions imposed on the Ricci tensor are in general neither over nor underdetermined, they give the most reasonable constraints one can place on the geometry. An Einstein metric g is one such that Ric(g) = cg, where c is a real constant.  This condition is a nonlinear system of partial differential equations on the underlying space. In General Relativity, the Ricci tensor of a Lorentz metric occurs as a term in Einstein's equation. Einstein metrics with Euclidean signature are important ingredients in supergravity theories such as string and M-theory. Understanding Einstein metrics has for decades been a central endeavour in Differential Geometry with significant impact on Theoretical Physics. My proposal focuses on the existence, moduli, and geometric properties of Einstein metrics, where two metrics are identified if they differ by a diffeomorphism.  Special emphasis will be placed on Einstein metrics whose holonomy algebra is generic because this case is currently the least understood by geometers. For positive Einstein metrics (c > 0) I will investigate their stability properties and develop a variational approach for the existence problem, starting with the cohomogeneity one case.  Einstein metrics are also studied using the Ricci flow by which one evolves a metric in the direction of -2Ric(g). This leads to a flow equation which has many properties similar to those of heat flow. Perelman discovered two useful functionals for studying the Ricci flow. The critical points of these functionals consist of Einstein metrics and their generalizations--the gradient Ricci solitons. I will also investigate the existence, moduli, and geometric properties of these and related structures, particularly in the much less understood non-Kahler case. In studying the Ricci flow, singularities of space tend to form in finite time. To analyse their formation, blow-up models are constructed by a dilation and limiting process, resulting in non-collapsed ancient or eternal solutions of the Ricci flow. I plan to study the construction of such solutions on non-compact spaces, including the important special case of shrinking solitons.  The proposed research may lead to large classes of new Einstein spaces, Ricci solitons, and ancient solutions with a wide range of topological and geometric properties. Some of the examples, especially explicit ones, may be useful to physicists as background geometries for sigma models or models in supergravity theories. The techniques developed may be useful for handling similar equations arising in geometry or physics.

黎曼度量是一种数学装置，允许我们计算 n 维空间中切向量之间的长度和角度。  黎曼曲率张量是从度量导出的基本对象，用于将其几何形状与平坦空间的几何形状进行局部比较。黎曼曲率张量中有一部分与度量 g 具有相同的数学类型——里奇张量 Ric(g)。由于施加在 Ricci 张量上的条件通常既不是超定的也不是欠定的，因此它们给出了人们可以对几何形状施加的最合理的约束。爱因斯坦度量 g 满足 Ric(g) = cg，其中 c 是实常数。  该条件是基础空间上的偏微分方程的非线性系统。在广义相对论中，洛伦兹度量的里奇张量作为爱因斯坦方程中的一项出现。具有欧几里得签名的爱因斯坦度量是超引力理论（例如弦理论和 M 理论）的重要组成部分。几十年来，理解爱因斯坦度量一直是微分几何的核心工作，对理论物理学产生了重大影响。我的建议重点关注爱因斯坦度量的存在性、模量和几何性质，其中如果两个度量因微分同胚而不同，则它们被识别。  将特别强调爱因斯坦度量，其完整代数是通用的，因为这种情况目前是几何学家理解最少的。对于正爱因斯坦度量 (c > 0)，我将研究它们的稳定性特性并开发一种解决存在问题的变分方法，从同质性案例开始。  爱因斯坦度量也使用 Ricci 流进行研究，通过该流在 -2Ric(g) 方向上演化出度量。这导致流动方程具有许多与热流相似的性质。佩雷尔曼发现了两个对研究里奇流有用的泛函。这些泛函的关键点包括爱因斯坦度量及其概括——梯度里奇孤子。我还将研究这些结构和相关结构的存在性、模量和几何特性，特别是在不太了解的非卡勒情况下。在研究利玛窦流时，空间奇点往往会在有限的时间内形成。为了分析它们的形成，通过膨胀和限制过程构建了爆炸模型，从而产生了里奇流的非塌陷的古代或永恒解决方案。我计划研究在非紧空间上构建此类解决方案，包括收缩孤子的重要特例。  拟议的研究可能会产生大量新的爱因斯坦空间、里奇孤子以及具有广泛拓扑和几何性质的古代解决方案。其中一些例子，尤其是明确的例子，可能对物理学家有用，作为西格玛模型或超引力理论模型的背景几何。所开发的技术可用于处理几何或物理学中出现的类似方程。