New Approaches to Spacetime Rigidity and Splitting

We take a new approach to Lorentzian splitting geometry, revamping and generalizing the classical notion of "horosphere" from hyperbolic geometry. We begin with a broad definition of Lorentzian sphere, which, in particular, gives an achronal boundary. Using an achronal decomposition of Penrose, we define the achronal limit of a sequence of monotonic achronal boundaries, and then a horosphere as an achronal limit of spheres whose centers approach infinity. As achronal limits are themselves achronal boundaries, our horospheres are C0 hypersurfaces by construction. In particular, this resolves, in an elegant and geometric way, the poor regularity of the Busemann function in the Lorentzian setting. Moreover, we show that such horospheres exhibit intrinsic support mean convexity properties, and using the maximum principle of [2], several splitting results are given under the timelike convergence condition, including applications to a conjecture of R. Bartnik (inspired by S.-T. Yau), related to the rigidity of the Hawking-Penrose singularity theorems. In particular, we construct two concrete examples, the ray and Cauchy horospheres, and give a proof of the conjecture in terms of the latter, under the additional assumption that a certain ‘max-min’ condition hold on its base Cauchy surface. Finally, turning attention to spacetimes with positive cosmological constant, we develop a notion of limit mean convexity and corresponding "maximum principle" for achronal limits, and use these to prove a rigid singularity result for asymptotically de Sitter spacetimes.