Einstein Manifolds and Related Geometric Structures

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，用于局部比较其几何与平坦空间的几何。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）。由于施加在里奇张量上的条件通常既不是超定的，也不是欠定的，因此它们给出了可以对几何形状施加的最合理的约束。爱因斯坦度量g是这样的度量：Ric（g）= cg，其中c是一个真实的常数。 这个条件是一个非线性系统的偏微分方程的基础空间。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。具有欧几里得签名的爱因斯坦度规是超引力理论如弦理论和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，对理论物理产生了重大影响。我的提案重点关注爱因斯坦度量的存在性、模和几何性质，其中如果两个度量存在微分同胚，则识别它们。 特别强调将放在爱因斯坦度量的holonomy代数是通用的，因为这种情况下是目前最不了解的geometers。对于正的爱因斯坦度量（c > 0），我将研究它们的稳定性，并发展一个变分方法的存在性问题，从共齐性的一种情况开始。 爱因斯坦度量也研究了使用的Ricci流，其中一个演变的方向上的度量-2Ric（g）。这导致了一个流动方程，它有许多类似于热流的性质。佩雷尔曼发现了两个有用的泛函研究里奇流。这些泛函的临界点由Einstein度规及其推广--梯度Ricci孤子组成。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，特别是在不太了解的非Kahler情况下。在研究里奇流时，空间的奇点往往在有限时间内形成。为了分析它们的形成，爆破模型是由一个膨胀和限制过程，导致非崩溃的古代或永恒的解决方案的里奇流。我计划在非紧空间上研究这种解的构造，包括收缩孤子的重要特例。 拟议的研究可能会导致大类新的爱因斯坦空间，里奇孤子，和古老的解决方案，具有广泛的拓扑和几何性质。一些例子，特别是显式的例子，可能对物理学家有用，作为超引力理论中的西格玛模型或模型的背景几何。所开发的技术可能是有用的，用于处理类似的几何或物理方程。

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.