We establish a complete classification theorem for the topology and for the orbital type of the null generators of compact non-degenerate Cauchy horizons of time orientable smooth vacuum \(3+1\)-spacetimes. We show that one and only one of the following must hold: (i) all generators are closed, (ii) only two generators are closed and any other densely fills a two-torus, (iii) every generator densely fills a two-torus, or (iv) every generator densely fills the horizon. We then show that, respectively to (i)–(iv), the horizon’s manifold is either: (i’) a Seifert manifold, (ii’) a lens space, (iii’) a two-torus bundle over a circle, or, (iv’) a three-torus. All the four possibilities are known to arise in examples. In the last case, (iv), (iv’), we show in addition that the spacetime is indeed flat Kasner, thus settling a problem posed by Isenberg and Moncrief for ergodic horizons. The results of this article open the door for a full parameterization of the metrics of all vacuum spacetimes with a compact Cauchy horizon. The method of proof permits direct generalizations to higher dimensions.

我们为时间可定向的光滑真空\（3 + 1 \）的紧凑型非退化柯西层位的零生成器的拓扑结构和轨道类型的轨道类型建立了完整的分类定理-时空。我们证明以下条件之一和唯一条件必须成立：（i）所有发电机都关闭，（ii）只有两个发电机都关闭，并且其他任何密集地填充两个托特尔，（iii）每个发电机密集地填充两个托特尔，或（iv）每个生成器都密集地填充了地平线。然后，我们表明，对于（i）-（iv），地平线的流形分别是：（i'）塞弗特流形，（ii'）晶状体空间，（iii'）在圆上的两个托勒斯束，或（iv'）三重奏。已知在示例中会出现所有四种可能性。在最后一种情况（iv），（iv'）中，我们还显示时空确实是平坦的Kasner，从而解决了Isenberg和Moncrief对于遍历视野提出的问题。本文的结果为紧凑的柯西视界为所有真空时空的度量的完整参数化打开了大门。