Nonlinear Partial Differential Equations in Geometry and General Relativity

该提案在几何分析的三个不同领域内提出了广泛的项目。 第一个领域涉及到广义相对论中Cauchy问题的非线性胶合技术的应用。非线性胶合技术在过去的20年里在几何分析中发挥了核心作用。 直到最近，它们才作为广义相对论的有用工具被引入。 它的应用包括具有任意空间拓扑的渐近平坦真空时空的存在性和没有极大切片的真空时空的存在性。 PI将以许多重要的方式使用和应用这些强大的工具。第二组项目研究smoothSchr\“odinger映射的存在性和行为。从几何的角度来看，薛定谔流是最自然的色散方程，因为它是流形之间映射的Dirichlet能量的Hamilton流。 这项工作将建立一个桥梁之间的发达理论dispersiveequations和重要的技术，从几何分析。第三部分是PI关于欧氏空间中常平均曲率曲面的工作的继续。 局部结构的模空间ofall这样的表面与一个固定的拓扑先前制定了由PI和他的合著者。 CMC表面新的gluingtechniques的发展和应用，导致了我们描述这些模空间的全局结构的能力的重要进展。 PI将把这项工作向前推进，以便我们开始对这些基本的几何物体有更深入的了解。广义相对论是物理学理论，它构成了我们理解宇宙大尺度结构的基石，自20世纪初成立以来，它与微分几何和偏微分方程密切相关。与任何物理理论一样，它的动力学表述（柯西问题）是核心问题。PI对柯西问题的研究将发展和应用新的分析技术到广义相对论。这些问题的解决将使我们进一步了解如何通过初始数据集的数学模型的物理现象的长期目标。在解析上，薛定谔流是经典非线性薛定谔方程的推广，已经得到了广泛的研究。所提出的关于Schroedinger流的方案将促进几何色散系统的一个新领域的发展。 常平均曲率曲面是指在保持封闭体积不变的情况下，局部极小化其表面积的曲面。 肥皂泡是常平均曲率曲面的一个常见而重要的例子. 这些研究项目的意义在于PI将获得的特定结果的重要性，以及复杂技术的持续发展，这些技术使人们能够接近和理解表现出日益复杂现象的问题。这方面的研究大部分是跨学科的不同领域withinMathematics和数学和物理之间。 虽然已经取得了重要成果，但在这些领域还有很大的扩展潜力。

This proposal presents a broad array of projects within three distinctareas of geometric analysis. The first area concerns the application ofnonlinear gluing techniques to the Cauchy problem in General Relativity.Nonlinear gluing techniques have played a central role in geometricanalysis over the last 20 years. It is only recently that they have beenintroduced as a useful tool in General Relativity. Applications haveincluded the existence of asymptotically flat vacuum spacetimes witharbitrary spacial topology and those with no maximal slices. The PI willextend and apply these powerful tools in a number of important ways. Thesecond set of projects studies the existence and behavior of smoothSchr\"odinger maps. From a geometric point of view the Schr\"odinger flowis the most natural dispersive equation as it arises as the Hamiltonianflow of the Dirichlet energy for maps between manifolds. This work willestablish a bridge between the well-developed theory of dispersiveequations and important techniques from geometric analysis. The third areais a continuation of the PI's work on surfaces of constant mean curvature(CMC) in Euclidean 3-space. The local structure of the moduli space ofall such surfaces with a fixed topology was previously worked out by thePI and his co-authors. The development and application of new gluingtechniques for CMC surfaces has led to important advances in our abilityto describe the global structure of these moduli spaces. The PI willcarry this work significantly forward so that we may begin to obtain amore thorough understanding of these basic geometric objects. General Relativity is the physical theory which forms the cornerstone toour understanding of the large scale structure of the Universe, and hasbeen intimately intertwined with differential geometry and partialdifferential equations since its inception at the beginning of the 20thcentury. As with any physical theory, its dynamical formulation (theCauchy problem) is of central concern. The PI's research on the Cauchyproblem will develop and apply new analytic techniques to GeneralRelativity. The resolution of these problems will bring us further in thelong term goal of understanding how to model physical phenomena via themathematics of the initial data sets. Analytically, the Schroedinger flowis a generalization of the classical nonlinear Schroedinger equation,which has been intensively studied. The proposed program concerning theSchroedinger flow will foster the development of a new area of geometricdispersive systems. Surfaces of constant mean curvature arise naturallyas the surfaces which locally minimize their surface area whilemaintaining a fixed enclosed volume. Soap bubbles form a familiar andimportant class of examples of surfaces of constant mean curvature. Thesignificance of these research projects lies both in the importance of theparticular results which the PI will obtain and also in the continualdevelopment of sophisticated techniques which enable one to approach andunderstand problems which exhibit increasingly complex phenomena. Much ofthis research is interdisciplinary both between distinct areas withinMathematics and between Mathematics and Physics. While important resultshave already been obtained, there is great potential for expansion inthese areas.