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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740851
• 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工具，它允许我们计算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，用于将其几何与平面空间的几何进行局部比较。黎曼曲率张量有一部分与度量g具有相同的数学类型--Ricci张量Ric(G)。由于对Ricci张量施加的条件通常既不是过多的，也不是确定不足的，它们给出了人们可以对几何施加的最合理的限制。爱因斯坦度量g是这样的度量：Ric(G)=Cg，其中c是实常数。这个条件是基础空间上的一个非线性偏微分方程组。在广义相对论中，洛伦兹度规的Ricci张量作为爱因斯坦方程中的一个项出现。具有欧几里得签名的爱因斯坦度规是弦理论和M理论等超引力理论的重要组成部分。几十年来，理解爱因斯坦度量学一直是微分几何的核心工作，对理论物理产生了重大影响。我的建议集中于爱因斯坦度量的存在、模和几何性质，其中如果两个度量通过微分同胚不同而被识别，则将特别强调其完整代数是通用的爱因斯坦度量，因为这种情况目前是几何学家最不了解的。对于正爱因斯坦度量(c&gt；0)，我将研究它们的稳定性，并发展一个变分方法来解决存在性问题，从同齐性开始。我们还使用Ricci流来研究爱因斯坦度量，通过Ricci流，人们可以向-2Ric(G)的方向演化度量。这导致了一个流动方程，它具有许多类似于热流的性质。佩雷尔曼发现了两个用于研究利奇流动的有用泛函。这些泛函的临界点由爱因斯坦度规及其推广--梯度利奇孤子组成。我还将调查这些结构和相关结构的存在、模数和几何性质，特别是在了解得少得多的非Kahler情况下。在研究Ricci流时，空间奇点往往是在有限时间内形成的。为了分析它们的形成，通过膨胀和限制过程建立了爆炸模型，导致了Ricci流的古老或永恒的非坍塌解。我计划在非紧空间上研究这类解的构造，包括收缩孤子这一重要特例。建议的研究可能会导致大类新的爱因斯坦空间、利玛窦孤子和具有广泛的拓扑和几何性质的古解。其中一些例子，特别是明确的例子，对于物理学家来说可能是有用的，作为西格玛模型或超引力理论中的模型的背景几何。所开发的技术可能对处理几何学或物理学中出现的类似方程有用。