Hausdorff Closed Limits and Rigidity in Lorentzian Geometry

We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the ‘achronal limits’ introduced by the authors in Galloway and Vega (Ann Henri Poincaré 15(11):2241–2279, 2014). This, in turn, allows for a broad generalization of the notion of Lorentzian horosphere introduced in Galloway and Vega (2014). We prove a new rigidity result for such horospheres, which in a sense encodes various spacetime splitting results, including the basic Lorentzian splitting theorem. We use this to give a partial proof of the Bartnik splitting conjecture (Bartnik in Commun Math Phys 117(4):615–624, 1988), under a new condition involving past and future Cauchy horospheres, which is weaker than those considered in Galloway (Some rigidity results for spatially closed spacetimes. Mathematics of gravitation, part I (Warsaw, 1996), Banach Center Publications, vol 41, Polish Academy of Science, Warsaw, pp 21–34, 1996) and Galloway and Vega (2014). We close with some observations on spacetimes with spacelike causal boundary, including a rigidity result in the positive cosmological constant case.