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.

Riemannian度量是一种数学设备，它使我们能够在N维空间中计算切线向量之间的长度和角度。 Riemann曲率张量是从局部比较其几何形状与平面空间的度量进行比较的基本对象。 Riemann曲率张量的一部分与ricci Tensor ric（G）的数学类型相同。由于对Ricci张量的条件通常超出或不确定，因此它们给出了可以放在几何形状上的最合理的约束。爱因斯坦公制G是一种使ric（g）= cg，其中c是一个真正的常数。该条件是基础空间上部分微分方程的非线性系统。总体而言，洛伦兹度量的ricci张量是爱因斯坦方程中的一个术语。具有欧几里得签名的爱因斯坦指标是超级理解爱因斯坦指标的重要因素，在差异几何学中一直是一项中心努力，对理论物理学有重大影响。我的提议着重于爱因斯坦指标的存在，模量和几何特性，如果通过差异为差异，则确定了两个指标。特殊重点将放在爱因斯坦的指标上，其全体代数是通用的，因为目前，这种情况是从几何学上理解的最少。对于阳性爱因斯坦指标（C> 0），我将研究它们的稳定性，并为存在问题开发出一种变异方法，从同时性一例开始。还使用RICCI流程研究了爱因斯坦指标，通过该流程，人们以-2RIC（g）的方向将公制进化。这导致了一个流动方程，其具有许多类似于热流的属性。佩雷尔曼（Perelman）发现了两个有用的功能来研究RICCI流动。这些功能的关键点包括爱因斯坦指标及其概括 - 梯度RICCI固体。我还将研究这些及相关结构的存在，模量和几何特性，尤其是在鲜为人知的非Kahler案例中。在研究RICCI流动时，空间的奇异性往往会在有限的时间内形成。为了分析其形成，通过词典和限制过程构建了爆炸模型，从而导致了RICCI流的非远古或永恒的解决方案。我计划研究在非紧密空间上的这种解决方案的构建，包括缩小固体的重要特殊情况。拟议的研究可能会导致大量的新爱因斯坦空间，RICCI实体和具有广泛拓扑和几何特性的古代解决方案。一些示例，尤其是显式的示例，可能对物理学家作为Sigma模型的背景几何技术或超级重力理论中的模型有用。开发的技术可能对于处理几何或物理中产生的类似方程可能很有用。