Achronal Limits, Lorentzian Spheres, and Splitting

Gregory J Galloway, Carlos Vega

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

In the early 1980s 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 Andersson and Galloway (Adv Theor Math Phys 6:307–327, 2002), and Cai and Galloway (Adv Theor Math Phys 3:1769–1783, 2000).

Original languageEnglish (US)
Pages (from-to)2241-2279
Number of pages39
JournalAnnales Henri Poincare
Volume15
Issue number11
DOIs
StatePublished - Oct 15 2014

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics

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