We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent.

我们证明，任何在四维及更高维度上具有正宇宙学常数的解析真空时空，包含具有最大对称紧凑横截面的静态极值杀伤视界，必须与极值史瓦西德西特解或其近邻局部等距。地平线几何（Nariai 解决方案）。在四维中，这意味着这些解是唯一包含具有紧凑横截面（直至识别）的静态极值视界的解析真空时空。我们还考虑了四维极值双曲史瓦西反德西特解的类似唯一性问题，并表明如果上同调阻碍了紧双曲曲面上的拉普拉斯反德西特解，则它可以简化为拉普拉斯算子的谱问题。地平线不存在。