Global Existence of the Critical Einstein-Equivariant Wave Map System

数学的一个基本和有趣的方面是，它提供了一种一致的语言来表达我们对自然界中发生的现象的理解。一个例子是波动方程，它模拟了许多自然现象，如振动弦、声波、水波、电磁波和线性化的爱因斯坦广义相对论方程。波动方程的几何和非线性推广是波动映射方程。研究波动图的动机是，它们自然地出现在爱因斯坦的广义相对论方程中，并被证明是理解爱因斯坦方程非线性效应的工具。该计划的目的是证明与爱因斯坦广义相对论方程耦合的等变波映射的全局存在性。提出的证明技术是基于弗里德兰德的表示公式和文森特·蒙克利夫最近的改进。除了在非线性波动方程的研究中引入新技术外，该项目还对广义相对论中的一个基本问题产生了影响：爱因斯坦方程的全局规律性，从而对罗杰·彭罗斯的宇宙审查猜想产生了影响。

A fundamental and interesting aspect of mathematics is that it provides a consistent language for expressing our understanding of the phenomena occurring in nature. An example is the wave equation which models many natural phenomena like vibrating strings, acoustic waves, water waves, electromagnetic waves and linearized Einstein's equations of general relativity. A geometric and nonlinear generalization of the wave equation is the wave maps equation. The motivation to study wave maps is that they occur naturally in Einstein's equations of general relativity and are proven tools to understand the nonlinear effects of Einstein's equations.The aim of the proposed project is to prove global existence of equivariant wave maps coupled to Einsteins equations of general relativity. The proposed techniques for the proof are based on Friedlander's representation formula and its recent improvements by Vincent Moncrief. In addition to introducing new techniques in the study of nonlinear wave equations, the project has implications for one of the fundamental questions in general relativity: the global regularity of Einstein's equations and consequently the cosmic censorship conjectures of Roger Penrose.