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建立引力物理模型提供了一种美丽而非常精确的方法。它可以用来预测黑洞、中子星和普通恒星碰撞的观测结果（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是什么样子的。处理爱因斯坦理论的一个特别有用的方法是将其表述为一个初值问题：根据这个公式，为了构建用于引力物理建模的时空，首先要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（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。该提案所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；它还涉及确定当一个人接近时空的“奇异区域”（例如，在大爆炸附近）时解的一般行为。除了研究爱因斯坦方程解的行为外，该基金还支持研究几何热流方程的解，如里奇流和平均曲率流。这里，我们感兴趣的是拓扑空间和它们所能支持的曲率类型之间的数学关系。值得注意的是，一些用于研究几何热流解的技术也适用于研究爱因斯坦方程的解。该基金支持的具体项目包括：1)爱因斯坦约束方程的解：已经开发了两种方法来构建和研究约束的解：第一种方法，保形方法，对于恒定平均曲率（“CMC”）和真空或电真空爱因斯坦约束（非正宇宙常数）的近CMC解非常有效，但似乎存在重大问题。该基金支持研究这些问题——解的非存在性和非唯一性——在许多情况下，包括渐近欧几里德解（“AE”）和渐近双曲解（“AH”），以及闭流形上的解。第二种方法是粘合，它允许将约束的已知解连接起来产生新解。， n体初始数据集。如果要用AH初始数据产生渐近平坦时空，它必须是“无剪切”的；因此，这笔拨款支持开发粘合技术的工作，该技术允许在无限处连接一对无剪切AH解，从而产生新的无剪切AH解（具有单个渐近区域）。2)强烈的宇宙审查：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”所预测的最大时空发展中普遍存在的测地线不完备性是否通常伴随着时空曲率爆炸。已知具有有限曲率的爱因斯坦方程的测地线不完全解（允许在柯西视界上扩展）；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各种解决方案，比如Gowdy时空。在这些证明中，验证“AVTD”行为（奇异区域附近时间导数优于空间导数）是一个至关重要的工具。PI和合作者开发了奇异初值问题作为识别AVTD行为的一种方法，并提出使用它来寻找具有一个杀戮场的真空解和没有杀戮场的爱因斯坦标量解中的非解析AVTD解。其他支持的工作试图证明Kasner解的AVTD行为在具有两个杀戮域的解之间是稳定的。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学时空的扩展方向。有很好的证据表明强烈吸引“熵”行为。这项拨款支持验证和探索这种行为的工作。4) Kahler几何附近的Ricci流：对于某些偶维流形M， M上的Kahler几何集合形成了M上所有黎曼几何空间的子空间。里奇流解是否从卡勒几何集合外开始但渐近地接近它？PI和他的合作者正在努力证明，对于一类特定的几何图形来说，情况就是这样。5)几何热流中颈夹行为的稳定性：对于旋转对称几何和嵌入，Ricci流和平均曲率流的颈夹行为是很好的理解。数值和分析的证据都表明，这种行为是稳定的。PI和他的合作者提出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.