A spacetime endowed with a globally defined timelike Killing vector field admits a certain model of warped product, called the standard static spacetime, and, when the volume element is modified by a factor that depends on a smooth function (which is called density function), we say that this ambient is a weighted standard static spacetime. In such spacetimes, we study some aspects of the geometry of spacelike hypersurfaces through of drift Laplacian of two functions support naturally related to them. For such hypersurfaces, with some restrictions on density function and the geometry of the ambient spacetime, we begin by stating and showing some results of uniqueness and nonexistence, several of them not assuming that the hypersurface to be of constant weighted mean curvature. Versions of these results are given for entire Killing graphs, that is, graphs constructed over an integral leaf of the distribution of smooth vector fields orthogonal to timelike Killing vector field. Finally, for closed spacelike hypersurface immersed in a weighted standard static spacetime with constant weighted mean curvature, we study a notion of stability via the first eigenvalue of the drift Laplacian.

具有全局定义的类似时间的Killing向量场的时空允许一个特定的变形产品模型，称为标准静态时空，并且，当体积元素被依赖于平滑函数（称为密度函数）的因子修改时，我们说这个环境是加权的标准静态时空。在这样的时空中，我们通过两个函数支持的漂移拉普拉斯算子研究了类空超曲面几何的某些方面。对于这样的超曲面，在密度函数和周围时空的几何形状上有一些限制，我们首先说明并显示一些唯一性和不存在的结果，其中一些不假定超曲面具有恒定的加权平均曲率。这些结果的版本适用于整个Killing图，也就是在与时态Killing矢量场正交的平滑矢量场分布的完整叶上构建的图。最后，对于沉浸在具有恒定加权平均曲率的加权标准静态时空中的封闭类空超曲面，我们通过漂移拉普拉斯算子的第一特征值研究了稳定性的概念。