Entire spacelike hypersurfaces of constant mean curvature in Minkowski Space
Entire spacelike hypersurfaces of constant mean curvature in Minkowski Space
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DOI：
10.1007/bf01404755
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发表时间：
1980
影响因子：
3.1
通讯作者：
Andrejs E. Treibergs
中科院分区：
文献类型：
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作者：
Andrejs E. Treibergs
恒定平均曲率的完整类空间超曲面在广义相对论中变得越来越有趣。作为场方程柯西问题的初始曲面，它们的理解可能有助于研究引力波的传播和奇点的形成。我们给出了原型时空 Minkowski 空间中这些表面的分类方案。我们可以改进构造渐近于任意维明可夫斯基空间中光锥的任意C2扰动的常平均曲率超曲面的方法。这里处理的数学问题类似于欧几里得空间中常平均曲率超曲面的唯一性和存在性问题。与欧几里得情况相反，在欧几里得情况下，整个零平均曲率图的存在意味着该图仅对于维度 n< 8 是一个平面，[参见例如 1]，在闵可夫斯基情况下，结论对于所有维度都成立 [2]。然而，与欧几里得的情况不同，最大定义的图在平移之前是唯一的[3]，在明可夫斯基的情况下，对于整个恒定平均曲率超曲面，除了平移和等距之外，解仍然存在一些不确定性。我们确定这种不确定性的程度。这里概述的证明的详细信息可以在[4, 10]中找到。
Complete spacelike hypersurfaces of constant mean curvature are becoming increasingly interesting in general relativity. Serving as initial surfaces for the Cauchy problem for the field equations, their understanding might be useful in the study of the propogation of gravitational waves and the formation of singularities. We give a classification scheme for these surfaces in the prototypical spacetime, Minkowski space. We can refine the method to construct constant mean curvature hypersurfaces asymptotic to arbitrary C2 perturbations of the light cone in Minkowski space of any dimension.The mathematical question treated here is analogous to the uniqueness and existence questions for hypersurfaces of constant mean curvature in Euclidean space. In contrast to the Euclidean case, where the existence of an entire zero mean curvature graph implies that the graph is a plane only for dimensions n< 8,[see, eg 1], in the Minkowski case, the conclusion is true for all dimensions [2]. However, unlike the Euclidean case where a maximally defined graph is unique up to translation [3], in the Minkowski case, for the entire constant mean curvature hypersurfaces, there is still some indeterminacy of solutions beyond translation and isometry. We determine the extent of this indeterminacy. Details of the proofs sketched here may be found in [4, 10].