  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].
