Einstein Manifolds and Related Geometric Structures

黎曼度规是一种数学工具，它允许我们计算n维空间中切向量之间的长度和角度。黎曼曲率张量是由度规导出的基本对象，用于将其几何形状与平面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规g具有相同的数学类型——里奇张量Ric(g)。由于施加在里奇张量上的条件通常既不是过定也不是欠定，它们给出了人们可以在几何上放置的最合理的约束。爱因斯坦度规g使得Ric(g) = cg，其中c是一个实常数。这个条件是底层空间上的一个非线性偏微分方程组。在广义相对论中，洛伦兹度规的里奇张量作为爱因斯坦方程中的一项出现。具有欧几里得特征的爱因斯坦度量是弦理论和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。我的建议集中在爱因斯坦度量的存在性、模和几何性质上，如果两个度量因微分同构而不同，则可以识别它们。我们将特别强调爱因斯坦度量，它的完整代数是一般的，因为这种情况是目前几何学家最不了解的。对于正的爱因斯坦度量（c > 0），我将研究它们的稳定性，并从齐性的一种情况开始，发展一种存在性问题的变分方法。爱因斯坦度规也用里奇流来研究，通过里奇流，一个度规在-2Ric(g)的方向上演化。这导致了一个流动方程，它具有许多类似于热流的性质。佩雷尔曼发现了研究里奇流的两个有用的函数。这些泛函的临界点由爱因斯坦度量及其推广组成——梯度里奇孤子。我还将研究这些结构和相关结构的存在性、模量和几何性质，特别是在鲜为人知的非kahler情况下。在研究里奇流时，空间奇点往往在有限时间内形成。为了分析它们的形成，爆破模型是通过膨胀和极限过程建立的，从而得到利奇流的非坍缩古解或永恒解。我计划在非紧空间上研究这些解的构造，包括收缩孤子的重要特例。提出的研究可能会导致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老解。其中一些例子，尤其是明确的例子，可能对物理学家有用，可以作为sigma模型或超重力理论模型的背景几何。所开发的技术可能对处理几何或物理中出现的类似方程有用。

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.