We study some aspects of the geometry of complete spacelike hypersurfaces immersed into a pp-wave spacetime, namely, into a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying suitable versions of the classical Hopf and Stokes theorems and a criterion of parabolicity for complete Riemannian manifolds, we obtain sufficient conditions which guarantee that a complete spacelike hypersurface is either maximal, 1-maximal or totally geodesic. As a consequence of these results, we also establish some results of nonexistence concerning such spacelike hypersurfaces. Finally, considering constant mean curvature closed spacelike hypersurfaces immersed in a pp-wave spacetime, we study a notion of stability via the first nonzero eigenvalue of the Laplacian.

中文翻译：

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沉浸在 pp 波时空中的完整类空间超曲面

我们研究了浸入 pp 波时空的完整类空间超曲面几何的某些方面，即进入允许平行光状矢量场的连接洛伦兹流形。最初，通过应用经典 Hopf 和 Stokes 定理的合适版本以及完整黎曼流形的抛物线标准，我们获得了充分条件，以保证完整的类空间超曲面是最大的、1-最大的或完全测地线的。作为这些结果的结果，我们还建立了一些关于此类空间超曲面的不存在结果。最后，考虑沉浸在 pp 波时空中的恒定平均曲率封闭类空间超曲面，我们通过拉普拉斯算子的第一个非零特征值研究稳定性的概念。