Our purpose in this paper is to study some geometric properties of spacelike hypersurfaces immersed into a pp-wave spacetime, namely, a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold, we obtain sufficient conditions which guarantee that a complete noncompact spacelike hypersurface with polynomial volume growth is either totally geodesic, maximal or 1-maximal. As a consequence, we establish nonexistence results concerning such spacelike hypersurfaces. Next, using a weak form of the Omori–Yau maximum principle, we get uniqueness and nonexistence results for stochastically complete spacelike hypersurface with constant mean curvature. Finally, we establish the notion of spacelike mean curvature flow soliton in pp-wave spacetimes and we provide some geometric conditions that allow us to guarantee how close a complete spacelike mean curvature flow soliton is to a totally geodesic immersion.

我们在本文中的目的是研究沉浸在pp 波时空中的类空超曲面的一些几何特性，即，一个连接的洛伦兹流形允许一个平行的类光矢量场。最初，通过对完全非紧黎曼流形上的光滑函数应用一种新形式的最大值原理，我们获得了充分的条件来保证具有多项式体积增长的完全非紧致类空超曲面是完全测地线的、最大的或 1-最大的。因此，我们建立了关于此类空间超曲面的不存在结果。接下来，使用 Omori–Yau 极大值原理的弱形式，我们得到具有恒定平均曲率的随机完备类空超曲面的唯一性和不存在性结果。最后，