局部等Lipschitz性质在Lorentz距离族对Cauchy超曲面叶状结构上的失效
Failure of local equi-Lipschitzness for families of Lorentz distances to Cauchy surface foliations
AI总结：
本文证明Lorentz距离族到Cauchy时间函数水平集的局部等Lipschitz连续性一般不成立，并提出与Bartnik分裂猜想等价的猜想。
摘要：
在时空中，沿一条完备类时线到和来自点的Lorentz距离函数族在该线邻域内是局部等Lipschitz连续的。这是经典Lorentz分裂定理证明中的一个基本组成部分。我们证明，对于给定Cauchy时间函数的水平集，该性质在Lorentz距离族到和来自这些水平集时一般会失效。此外，我们基于宇宙学和类时测地完备时空中Cauchy时间函数的存在性，提出使得Lorentz距离到其水平集是等Lipschitz的猜想，这些猜想等价于Bartnik的分裂猜想。
英文摘要：
The families of Lorentzian distance functions to and from points along a complete timelike line in a spacetime are known to be locally equi-Lipschitz continuous in a neighborhood of the line. This is an essential component in the proof of the classical Lorentzian splitting theorems. We show that this property fails in general for families of Lorentzian distances to and from the level sets of a given Cauchy temporal function. Moreover, we formulate conjectures based on the existence of Cauchy temporal functions in cosmological and timelike geodesically complete spacetimes such that the Lorentz distances to its level sets are equi-Lipschitz, which are equivalent to Bartnik's splitting conjecture.
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