Achronal limits, Lorentzian spheres, and splitting

TL;DR

This paper introduces a new geometric approach to Lorentzian horospheres using achronal limits, leading to regularity results and a splitting theorem that address longstanding problems in spacetime geometry and singularity theorems.

Contribution

It generalizes the concept of horospheres in Lorentzian geometry via achronal limits and establishes a splitting theorem with applications to the Bartnik conjecture and de Sitter spacetimes.

Findings

Established a splitting theorem for generalized horospheres.

Derived new results on the Bartnik splitting conjecture.

Obtained a rigid singularity result for asymptotically de Sitter spacetimes.

Abstract

In the early 80's S.-T. Yau posed the problem of establishing the rigidity of the Hawking-Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets - the horospheres. The regularity theory in the Lorentzian case is considerably more complicated and less complete than in the Riemannian case. In this paper we introduce a broad generalization of the notion of horosphere in Lorentzian geometry and take a completely different (and highly geometric) approach to regularity. These generalized horospheres are defined in terms of 'achronal limits', and the improved regularity we obtain is based on regularity properties of achronal boundaries. We establish a splitting result for generalized horospheres, which when specialized to 'Cauchy horospheres' yields new results on the Bartnik splitting conjecture, a concrete realization of the problem posed by Yau. Our methods are also applied to spacetimes with positive cosmological constant. We obtain a rigid singularity result for future asymptotically de Sitter spacetimes related to results in [1,7].