On the Behavior of Solutions of Einstein's Equations and Solutions of Geometric Heat Flow Systems

爱因斯坦的引力场理论为在天体物理和宇宙尺度上模拟引力物理提供了一种美丽而极其准确的方法。它可以用来预测黑洞、中子星和普通恒星碰撞的观测后果(包括电磁辐射和引力辐射)，也可以用来从观测中辨别大爆炸的样子。使用爱因斯坦理论的一种特别有用的方法是将其描述为一个初值问题：根据这个公式，为了构建用于建模引力物理的时空，人们首先为时空选择一个初始状态(在某个感兴趣的任意时间)，然后通过演化到这个初始状态的过去和未来来构建时空。爱因斯坦的方程式(爱因斯坦理论的核心)既控制了初始状态的可能选择，也决定了进化如何进行。这项提议支持的研究涉及如何选择满足爱因斯坦约束方程的初始状态；它还涉及当一个人接近时空的“奇异区域”(例如，大爆炸附近)时，确定解的一般行为。除了研究爱因斯坦方程的解的性态外，这笔赠款还支持几何热流方程的解的研究，如Ricci流和平均曲率流。这里，感兴趣的是拓扑空间和它们可以支持的曲率类型之间的数学关系。值得注意的是，几何热流解的研究中使用的一些技术在研究爱因斯坦方程的解时也很有用。这项资助支持的具体项目如下：1)爱因斯坦约束方程的解：已经开发了两种方法来构造和研究约束的解：第一种方法，共形方法，非常适合于常平均曲率(CMC)和近CMC解的真空或电的爱因斯坦约束(具有非正的宇宙常数)，但似乎有其他主要问题。这笔赠款支持研究这些问题的工作-解的不存在和不唯一-在许多情况下，包括渐近欧几里得(“AE”)和渐近双曲(“AH”)解，以及闭流形上的解。第二种方法，粘合，允许连接约束的已知解决方案以产生新的约束--例如，N体初始数据集。AH初始数据如果要用来产生渐近平坦的时空，就必须是“无切变的”；因此，这笔赠款支持开发粘合技术，允许一对无切变的AH解在无穷远处连接，从而产生一个新的无切变的AH解(只有一个渐近区域)。2)强宇宙审查：近50年来，数学相对论中的一个主要问题是，霍金-彭罗斯“奇点定理”预测的最大时空发展中普遍存在的测地线不完全性是否通常伴随着时空曲率的膨胀。有界曲率的爱因斯坦方程的大地测量不完全解(允许在柯西地平线上延伸)是已知的；但“强宇宙审查”(“SCC”)猜想表明，这种情况并不普遍发生。SCC的模型版本已经被证明适用于一系列解决方案，例如Gowdy时空。在这些证明中，验证“AVTD”行为(时间导数在奇点区域附近相对于空间导数的优势)一直是一个至关重要的工具。PI和合作者发展了奇异初值问题作为识别AVTD行为的一种方法，并建议使用它在具有一个杀伤场的真空解和没有杀伤场的爱因斯坦标量解之间寻找非解析的AVTD解。其他支持的工作试图证明Kasner解的AVTD行为在具有两个杀伤场的解中是稳定的。3)扩展宇宙学：PI和他的合作者建议使用数值和分析研究相结合的方法来探索模型宇宙时空的扩展方向。有很好的证据可以很好地吸引“熵”行为。这笔赠款支持验证和探索这一行为的工作。4)Kahler几何附近的Ricci流：对于某些偶数维流形M，M上的Kahler几何集构成M上所有黎曼几何空间的子空间。是否存在开始于Kahler几何集合之外但渐近于该集合的Ricci流解？PI和他的合作者正在努力证明，对于特定的几何类型，情况就是这样。5)几何热流中Neckpinch行为的稳定性：对于旋转对称的几何和嵌入，Ricci流和平均曲率流中的Neckpinch行为是众所周知的。无论是数字上的还是分析上的，都有证据表明这种行为是稳定的。PI和他的合作者提出了进一步的工作来验证这种稳定性。

Einstein's gravitational field theory provides a beautiful and remarkably accurate means for modeling gravitational physics on both the astrophysical and cosmological scales. It can be used to predict the observational consequences (both in terms of electromagnetic and gravitational radiation) of black holes and neutron stars and ordinary stars colliding, and it can also be used to discern from observations what the Big Bang was like. One particularly useful way to work with Einstein's theory is by formulating it as an initial value problem: According to this formulation, to construct spacetimes of use in modeling gravitational physics, one first chooses an initial state for the spacetime (at some arbitrary time of interest), and one then constructs the spacetime by evolving both into the past and the future of this initial state. Einstein's equations (at the heart of Einstein's theory) both control possible choices of the initial state, and determine how the evolution proceeds. The research supported in this proposal involves how to make choices of the initial state which satisfy the Einstein constraint equations; it also involves determining the generic behavior of solutions as one approaches "singular regions" of the spacetime (near the Big Bang, for example). Besides studies of the behavior of solutions of Einstein's equations, this grant also supports studies of solutions of geometric heat flow equations such as the Ricci flow and mean curvature flow. Here, the interest is in the mathematical relationship between topological spaces and the types of curvature that they can support. Remarkably, some of the techniques used in the study of geometric heat flow solutions are also useful in studying solutions of Einstein's equations.Among the specific projects supported by this grant are the following:1) Solutions of the Einstein constraint equations: Two approaches have been developed for constructing and studying solutions of the constraints: The first of these, the conformal method, works beautifully for constant mean curvature ("CMC") and near-CMC solutions of the vacuum or electrovac Einstein constraints (with nonpositive cosmological constant), but appears to have major problems otherwise. This grant supports work which studies these problems---non-existence and non-uniqueness of solutions---in a number of cases, including asymptotically Euclidean ("AE") and asymptotically hyperbolic ("AH") solutions, as well as solutions on closed manifolds. The second approach, gluing, allows known solutions of the constraints to be joined to produce new ones--e.g., N-body initial data sets. AH initial data must be "shear-free" if it is to be used to produce asymptotically flat spacetimes; hence this grant supports work to develop gluing techniques which allow the joining at infinity of a pair of shear-free AH solutions, thereby producing a new shear-free AH solution (with a single asymptotic region).2) Strong Cosmic Censorship: For almost 50 years, one of the major questions in mathematical relativity has been if the ubiquitous geodesic incompleteness in maximal spacetime developments predicted by the Hawking-Penrose "singularity theorems" is generically accompanied by spacetime curvature blowup. Geodesically incomplete solutions of Einstein's equations with bounded curvature (allowing extensions across a Cauchy horizon) are known; but the "Strong Cosmic Censorship ("SCC") conjecture suggests that this does not happen generically. Model versions of SCC have been proven for families of solutions, such as the Gowdy spacetimes. In these proofs, verifying "AVTD" behavior (dominance of time derivatives over space derivatives near the singular region) has been a crucial tool. The PI and collaborators has developed the singular initial value problem as a way of identifying AVTD behavior, and proposes to use it to find non-analytic AVTD solutions among vacuum solutions with one Killing field, and among Einstein-scalar solutions with no Killing fields. Other supported works seeks to show that the AVTD behavior of Kasner solutions is stable among solutions with two Killing fields. 3) Expanding Cosmologies: The PI and his collaborators propose to use a combination of numerical and analytical studies to explore the expanding direction of model cosmological spacetimes. There is good evidence for strongly attracting "entropic" behavior. This grant supports work to verify and explore this behavior. 4) Ricci Flow Near Kahler Geometries: For certain even-dimensional manifolds M, the set of Kahler geometries on M forms a subspace of the space of all Riemannian geometries on M. A Ricci flow solution which begins at a Kahler geometry remains Kahler. Are there Ricci flow solutions which begin outside the set of Kahler geometries but asymptotically approach it? The PI and his collaborators are working to show that this is the case, for a certain class of geometries.5) Stability of Neckpinch Behavior in Geometric Heat Flows: Neckpinch behavior in Ricci flow and mean curvature flow is well-understood for rotationally symmetric geometries and embeddings. There is evidence, both numerical and analytical, that such behavior is stable. The PI and his collaborators propose further work to verify this stability.