真空初值数据集空间的拓扑性质
On the topology of the space of vacuum initial data sets
- Université de Tours（图尔大学）
- KTH Stockholm（斯德哥尔摩皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文通过参数化共形方法，证明闭流形上真空初值数据集空间具有非平凡同伦群，并首次揭示其全局拓扑性质。
中文摘要 AI 辅助
我们证明了闭流形上的真空初值数据集空间通常具有许多非平凡的同伦群。出发点是第二作者的一个结果，该结果在满足严格主能量条件的初值数据集空间的同伦群中构造了非平凡元素，以及与Bernd Ammann合作的后续结果表明，当去掉严格性假设时，这些元素通常仍然存在。在本工作中，我们使用共形方法的参数化版本，证明这些元素也可以由映射到真空初值数据集空间来表示。这需要两个可能独立有趣的结果：允许共形 Killing 向量的度量可以从度量空间中移除而不改变其弱同伦类型，并且在剩余的度量上，York 分解可以在族中执行，TT-张量形成一个平凡的 Hilbert 丛。据我们所知，这是关于该空间全局拓扑的第一个结果。
英文摘要
We show that the space of vacuum initial data sets on a closed manifold often has many non-trivial homotopy groups. The starting point is a result of the second named author, which constructs non-trivial elements in the homotopy groups of the space of initial data sets satisfying the strict dominant energy condition, together with a later result in joint work with Bernd Ammann showing that these elements often persist when the strictness assumption is dropped. In this work, we use a parametrized version of the conformal method to show that these elements may also be represented by maps into the space of vacuum initial data sets. This requires two results that may be of independent interest: metrics admitting conformal Killing vectors can be removed from the space of metrics without changing its weak homotopy type, and over the remaining metrics the York decomposition can be carried out in families, the TT-tensors forming a trivial Hilbert bundle. To our knowledge, this is the first result on the global topology of this space.
