渐进平坦流形上的耦合曲率流

Asymtotically flat manifolds were arises from the study of the Isolated gravitating system, and it is an important object in General relativity as well as the differential geometry. Geometric flow is to study the evolution equation of the geometric quantities on Manifolds. One of these flows is the Ricci flow, which was successfully used to solve the Poincare conjecture as well as the Differentiable sphere theorem. On the other hand, the extrinsic flow such as mean curvature flow was also studied. In this project, we consider the coupled flow such as the mean curvature flow inside ambient manifold evolving by normalized Ricci flow. By choosing different extrinsic flow and the evolving method of the ambient manifold, this coupled flow will have applications on the mass related problem of Asymptotically flat manifold in General relativity.

渐进平坦流形产生于孤立引理系统的研究，是广义相对论中非常重要的概念，也是一种重要微分几何研究对象。几何曲率流研究的是流形的几何量按某种方式发展演化从而改变流形结构和性质的过程。其中，Ricci流最近被成功地用于解决几何化猜想、庞加莱猜想和微分球面定理。与此同时，以平均曲率流为代表的外蕴曲率流也被广泛地研究，并且在微分几何和物理等方面都找到了广泛的应用。本项目研究外蕴和内蕴耦合的曲率流，例如外围空间按Ricci流演化的平均曲率流。根据问题需要，我们可以选择不同的外蕴曲率流和外围流形的演化方式。我们试图研究这种这种耦合流的收敛性。同时我们将用它来研究广义相对论里中的质量的问题，比如渐进平坦流形的质心问题，正质量定理等等。希望能给这类问题以一个新的视角。