论文标题
带有完整纤维和calabi-bernstein型问题的扭曲产品空间中的间距性超曲面
Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi-Bernstein-type problems
论文作者
论文摘要
在本文中，研究了带有完整纤维的扭曲产品空位的空格超曲面。有几种条件确保呈现全球双曲线，以及需要在$ i \ i \ times_f f $中保留每个空格表情的关系，以使其成为一种简单的扭曲产品。当假定纤维被闭合（紧凑而没有边界）并且环境时空具有合适的扩展行为时，可获得恒定平均曲率超出曲面的不存在结果。在相同的假设下，提出了紧凑的最大超曲面和其他完全脐带的表征，对其平均曲率有适当的限制。还包括一维Lorentzian Fiber的最大超出曲面的描述。最后，计算了光纤上的空格图的平均曲率方程，作为应用，一些calabi-Bernstein-type结果得到了证明。我们还将一些已知的共形几何结果包括在附录中，这些结果描述了相关张量和操作员在伪里曼尼亚背景中的共形图的作用下的转换。
In this article spacelike hypersurfaces immersed in twisted product spacetimes $I\times_f F$ with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, as well as a relation that needs to hold on each spacelike hypersurface in $I\times_f F$ for it to be a simple warped product. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersurfaces and other for totally umbilic ones with a suitable restriction on their mean curvature are presented. The description of maximal hypersurfaces in twisted product spacetimes of the form $I\,{ }_{f}\!\!\times F$ with a one-dimensional Lorentzian fiber is also included. Finally, the mean curvature equation for a spacelike graph on the fiber is computed and as an application, some Calabi-Bernstein-type results are proven. We also include in an Appendix some known conformal geometry results describing the transformation of relevant tensors and operators under the action of a conformal map in a pseudo-Riemannian background.
