论文标题
静态近地平线几何形状和准元素歧管的刚性
Static near horizon geometries and rigidity of quasi-Einstein manifolds
论文作者
论文摘要
地平线几何形状附近的静态真空是封闭的歧管$ m $上的某个准伊因斯坦方程的解决方案$（m，g，x），其中$ g $是Riemannian度量标准，$ x $是封闭的1型。众所周知，当宇宙常数消失时，就会存在刚性：$ x $消失，因此$ g $是ricci平坦的。我们研究了宇宙常数的所有迹象的刚性形式。有人断言，当宇宙常数为负时，这种刚度也会成立，但我们表现出反例。我们表明，对于负宇宙常数，如果$ x $不消失相同，它必须是不可压缩的，具有恒定的规范，并且在共同体中是不繁琐的，并且$（m，g）$必须具有恒定的标量曲率和零欧拉的特征。如果宇宙常数为正，则$ x $必须确切（如果$ \ dim m = 2 $）消失。我们的结果更普遍地适用于封闭歧管上的一系列准元素方程。我们向封闭的$ x $案例扩展了一些准确的1型$ x $的准伊因斯坦指标的已知结果。我们考虑对真空条件放松的近地平线几何形状，以允许存在有限的物质场。附录包含了卢西蒂（Lucietti）对Yamabe类型的准伊因斯坦紧凑型指标（带有任意$ x $）的概括。
Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: $X$ vanishes and consequently $g$ is Ricci flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if $X$ does not vanish identically, it must be incompressible, have constant norm, and be nontrivial in cohomology, and $(M,g)$ must have constant scalar curvature and zero Euler characteristic. If the cosmological constant is positive, $X$ must be exact (and vanishing if $\dim M=2$). Our results apply more generally to a broad class of quasi-Einstein equations on closed manifolds. We extend some known results for quasi-Einstein metrics with exact 1-form $X$ to the closed $X$ case. We consider near horizon geometries for which the vacuum condition is relaxed somewhat to allow for the presence of a limited class of matter fields. An appendix contains a generalization of a result of Lucietti on the Yamabe type of quasi-Einstein compact metrics (with arbitrary $X$).
