On the Behavior of Solutions of Einstein's Equations and Other Geometric Partial Differential Equations

除了为我们提供了一个令人惊讶的精确和美丽的模型，用于研究天体物理和宇宙尺度上的引力物理学，爱因斯坦？的引力场理论是数学上有趣问题的丰富来源。我们最感兴趣的一些问题与爱因斯坦约束方程的解有关。这些方程限制了人们对演化引力系统的初始数据的选择。我们感兴趣的是参数化约束的解的集合（即，寻找自由度），开发用于构造解的算法，以及研究约束的解的行为。我们希望能运用？胶合？来实现爱因斯坦约束。胶合的思想是，给定约束的一对解，人们试图通过连接每个解中的一个点的桥将两个解平滑地连接起来。我们已经应用了粘合技术的爱因斯坦约束，并已使用它来构建多黑洞的初始数据集，添加虫洞给定的解决方案，构建初始数据的黑洞在宇宙学的解决方案，并表明，任何给定的封闭流形（减去一个点）承认一个渐近平坦的解决方案的约束。到目前为止，我们在胶合方面的工作都假设给定的解具有恒定平均曲率的区域，并且是真空解。我们正在努力消除这些假设，这应该会开辟更广泛的应用范围。

In addition to providing us with an amazingly accurate and beautiful model for studying gravitational physics on both the astrophysical and the cosmological scale, Einstein?s gravitational field theory is a rich source of mathematically interesting questions. A number of the questions we are most interested in pertain to solutions of the Einstein constraint equations. These equations restrict the choice of initial data one can make for an evolving gravitational system. We are interested in parametrizing the set of solutions of the constraints (i.e., finding the degrees of freedom), developing algorithms for constructing solutions, and studying the behavior of solutions of the constraints. We hope to apply the technique of ?gluing? to implement the Einstein constraints. The idea of gluing is that, given a pair of solutions of the constraints, one attempts to connect the two solutions smoothly via a bridge joining a point in each of them. We have applied the gluing technique to the Einstein constraints, and have used it to construct multi-black hole initial data sets, to add wormholes to given solutions, to construct initial data for black holes in cosmological solutions, and to show that any given closed manifold (minus a point) admits an asymptotically flat solution of the constraints. Our work in gluing so far has assumed that the given solutions have a region of constant mean curvature, and are vacuum solutions. We are working to remove these assumptions, which should open up a much wider range of applications.