时空中的常平均曲率超曲面

As recent cosmological observations indicated that our universe undergo accelerated expansion, asymptotically de Sitter spacetime gains much importance. Unlike the case for asymptotically flat nor asymptotically anti-de Sitter spacetime, positive energy theorem for asymptotically de Sitter spacetime holds true only for those initial data sets, whose mean curvatures are bounded by some certain constant. Thus an important question comes up: When do the asymptotically de Sitter spacetime possess some certain constant mean curvature spacelike hypersurfaces? This project mainly aims at this question. We hope to make clear the interaction between the existence of the constant mean curvature hypersurfaces, and the geometric features of their background asymptotically de Sitter spacetime. These researches will help to reveal the underlying geometric features of the gravity of asymptotically de Sitter spacetime.

天文观测数据表明，我们所处的宇宙正加速膨胀。因而，对于渐近de Sitter时空的理论研究受到重视。与渐近平坦或渐近anti-de Sitter的时空不同，渐近de Sitter时空的正能量定理只有当初始资料集满足一定的平均曲率条件时才成立。从而一个基本的问题就是，渐近de Sitter时空何时存在特定常平均曲率的类空超曲面。这构成本项目的主要研究内容。我们希望了解常平均曲率超曲面的存在，与其所在的渐近de Sitter时空的几何性质是如何相互反映。这将有助于进一步理解渐近de Sitter时空的引力特性的几何本质。

天文观测发现宇宙在加速膨胀，使得正宇宙时空的研究受到关注。本项目主要研究渐近de Sitter时空中一类常平均曲率非紧类空超曲面的存在性。我们证明了标准de Sitter时空的planar坐标系下，对时空度量作整体的微扰，则平均曲率为常数$\frac{3}{\lambda}$的非紧类空超曲面存在，且渐近于时间面。我们还简单讨论该常平均曲率曲面上的能量，及其和时间面上能量的关系。