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首页> 外文期刊>Annales Henri Poincare >Constraint Equations for 3+1 Vacuum Einstein Equations with a Translational Space-Like Killing Field in the Asymptotically Flat Case
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Constraint Equations for 3+1 Vacuum Einstein Equations with a Translational Space-Like Killing Field in the Asymptotically Flat Case

机译：渐近平坦情况下具有平移类空间杀伤场的3 + 1真空爱因斯坦方程的约束方程

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摘要

We solve the Einstein constraint equations for a 3 + 1-dimensional vacuum space-time with a space-like translational Killing field. The presence of a space-like translational Killing field allows for a reduction of the 3 + 1-dimensional problem to a 2 + 1-dimensional one. Vacuum Einstein equations with a space-like translational Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case. In the case where an additional rotational symmetry is added, the problem has a long history. In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on R-2. The main difficulty in that case is due to the delicate inversion of the Laplacian on R-2. In particular, we have to work in the non-constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.
机译：我们用一个类似空间的平移Killing场来求解3 +1维真空时空的爱因斯坦约束方程。像空间一样的平移Killing场的存在使3 +1维问题减少到2 +1维问题。在紧凑的情况下，Choquet-Bruhat和Moncrief研究了具有空间平移Killing场的Vacuum Einstein方程。在添加附加的旋转对称性的情况下，该问题具有悠久的历史。在本文中，我们考虑渐近平坦的情况。这对应于在R-2上求解非线性椭圆系统。在这种情况下的主要困难是由于拉普拉斯算子在R-2上的精细反转。特别是，我们必须在非恒定平均曲率设置中进行工作，这迫使我们考虑爱因斯坦约束方程的复杂耦合。

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