Splitting theorems for hypersurfaces in Lorentzian manifolds

The splitting problem for spacetimes with timelike Ricci curvature bounded below by zero has been discussed extensively in the past (most notably by Eschenburg, Galloway and Newman), in particular there exist versions for both spacetimes containing a complete timelike line and spacetimes containing a maximal hypersurface $\Sigma$ and a (future) complete $\Sigma$‑ray. For timelike Ricci curvature bounded below by some $\kappa \gt 0$ only an analogue to the first case has been shown explicitly (see [AGH96]).

In this paper we employ their methods (a geometric maximum principle for the level sets of the Busemann function) to study analogues of the second case for hypersurfaces with mean curvature bounded from above by $\beta$. We show that given a $\Sigma$‑ray of maximal length $J^{+}(\Sigma)$ is isometric to a warped product if either $\kappa \gt 0$ or $\beta \leq -(n-1) \sqrt{\lvert \kappa \rvert}$. Additionally we present an elementary proof for such a splitting if one assumes that the volume of (future) distance balls over subsets of this hypersurface is maximal.

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The author is the recipient of a DOC Fellowship of the Austrian Academy of Sciences at the Institute of Mathematics at the University of Vienna. This work was also partially supported by the Austrian Science Fund (FWF) project number P28770.

Received 5 December 2016

Accepted 23 October 2017

Published 12 March 2020