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

该奖项支持的研究涵盖了广义相对论、引力物理和几何分析等领域的广泛项目。许多项目都与爱因斯坦的引力场方程告诉我们的黑洞性质有关。我们希望完成一个证明，在任何时空维度中，处于平衡状态的黑洞必须具有空间对称性。我们还计划继续研究非平衡状态的天体物理模型系统：什么时候能识别出这样一个系统是未来的黑洞，在拓扑封闭的宇宙中能做到这一点吗？我们提出的其他项目包括在数学上探索大爆炸附近引力场的本质：是否在大爆炸附近通常会发现曲率（潮汐）奇点，以及目前在宇宙模型的受限族中看到的振荡行为的支持是否实际上扩展到不那么受限的族？除了这些涉及广义相对论的项目外，我们提出的研究还包括对利玛窦流的研究。这个偏微分方程组最近在庞加莱猜想和几何化猜想的证明中发挥了重要作用，是进一步研究几何与拓扑关系的一个很有前途的工具。我们已经成功地研究了里奇流中某些平衡几何形状的稳定性，我们计划将这项工作扩展到更广泛的类别（包括坍缩几何形状）。

The research supported by this award covers a wide variety of projects in general relativity, gravitational physics, and geometric analysis. A number of the projects are concerned with what Einstein's equations for the gravitational field tell us about properties of black holes. We hope to complete a proof that, in any spacetime dimension, black holes in equilibrium must have spatial symmetries. We also plan to continue a program of study of model astrophysical systems which are not in equilibrium: when can one identify such a system as a future black hole, and can one do this in a topologically closed universe? Other projects we are proposing involve mathematically probing the nature of gravitational fields near the Big Bang: is a curvature (tidal) singularity generally to be found near the Big Bang, and does the current support for oscillatory behavior seen in restricted families of cosmological models in fact extend to less restricted families?In addition to these projects involving general relativity, our proposed research includes studies of Ricci flow. This system of partial differential equations, which has recently played a major role in the proof of the Poincare and the geometrization conjectures, is a promising tool for further studies of the relationship between geometry and topology. We have had success in studying the stability of certain equilibrium geometries in Ricci flow, and we plan to extend this work to a wider class (including collapsing geometries.)