Global existence problems for the Einstein equations

DMS-0104402 Lars Andersson这个提议涉及爱因斯坦方程的整体存在性和宇宙监督问题，以及洛伦兹几何和黎曼几何中的相关问题。主要目标之一是证明在宇宙学情况下爱因斯坦方程的非线性稳定性结果。对于大数据，全局存在性是一个非常困难的问题，这里的目的是识别和研究模型问题以及由对称约化导出的低维问题。宇宙学奇点的结构将从BKL建议的观点来研究。 对于平坦的、空间紧凑的时空，将研究全局常平均曲率叶化，特别是它们在膨胀方向和奇点附近的渐近行为。爱因斯坦方程描述了物质与时空几何的相互作用，是现代物理学的基本经典场方程之一。根据现代物理学的世界观，我们的宇宙在大尺度上可以用爱因斯坦方程的最简单解之一--弗里德曼-勒梅特解来描述。根据这幅图，宇宙是从一个奇点（大爆炸）发展而来的，并且正在膨胀。该项目的工作涉及该模型的非线性稳定性，大爆炸附近极端条件下时空的行为及其在膨胀方向上的最终命运。

DMS-0104402Lars Andersson This proposal is concerned with the global existence and cosmic censorship problems for the Einstein equation, and related issues in Lorentzian and Riemannian geometry. One of the primary goals is to prove a nonlinear stability result for the Einstein equation in the cosmological case. For large data, global existence is a very hard problem, and here the aim is to identify and study model problems as well as lower dimensional problems derived by symmetry reductions. The structure of cosmological singularities will be studied from the point of view of the BKL proposal. For flat, spatially compact spacetimes, global constant mean curvaturefoliations will be studied, in particular their asymptotic behavior in the expanding direction and near the singularity.The Einstein equation, which describes the interaction of matter with spacetime geometry, is one of the fundamental classical field equations of modern physics. According to the world view of modern physics, our universe is described on large scales by one of the simplest solutions to the Einstein equations, the Friedman-Lemaitre solution. According to thispicture, the universe has developed from a singularity (big bang) and is expanding. The work in this project is concerned with the nonlinear stability of this model, the behavior of spacetime in extreme conditions near the big bang and its ultimate fate in the expanding direction.