by Piotr T. Chruściel and Gregory J. Galloway
Introduction
Indeed, Yvonne has written several papers concerning existence and uniqueness properties of maximal (and CMC) hypersurfaces in spacetime manifolds. In [1] she states, “The existence of a maximal submanifold (with respect to area) is an important property for a space time, hyperbolic riemannian manifold satisfying Einstein equations.” She goes on to describe the importance of this for solving the Einstein constraint equations, and also the relevance of this (at the time) for proving positivity of mass.
Meanwhile the existence of maximal hypersurfaces has been found to be useful for other purposes as well. Of particular relevance here is a classical result of Bartnik [e6] that establishes, under certain conditions, the existence of maximal hypersurfaces in asymptotically flat spacetimes. This and related results have been useful for a number of purposes in the study of such spacetimes. Given the role of asymptotically anti-de Sitter spacetimes in theoretical physics (e.g., via the AdS/CFT correspondence), one is naturally led to consider maximal hypersurfaces in this setting, where now the notion of renormalised volume also becomes relevant.
The question then arises of existence of maximal hypersurfaces in \( (n{+}1) \)-dimensional, \( n\ge 2 \), asymptotically locally hyperbolic spacetimes, with controlled asymptotic behaviour at the conformal boundary at infinity so that the renormalised volume is defined. The simplest case where an affirmative answer can be given is in a perturbative setting, when the metric, or the asymptotic data, or both, are perturbed away from a maximal slicing, and when an energy condition is satisfied. We prove this in Theorem 9.3 below, generalising a related result of [e14].
However, simple examples show that some global regularity conditions need to be satisfied by the spacetime for existence in general. Here two conditions arise naturally: that of existence of barriers, and that of compactness of the domain of dependence of a fiducial Cauchy surface. Under these two conditions we show solvability of an asymptotic Dirichlet problem for the maximal hypersurface equation; see Theorems 9.1 and 9.5. This generalises previous results of [e7], [e15], established under restrictive conditions on the dimension or on the class of spacetimes considered.
As such, these last two theorems do not guarantee that a well-defined notion of renormalised volume exists without further conditions; see for instance Theorem 9.2 where both “good barriers” and timelike convergence is assumed. This is related to the question of the behaviour of the maximal hypersurface at the conformal boundary at infinity, which turns out to be delicate. Indeed, while maximal hypersurfaces are typically spacelike and smooth in the interior, they might become asymptotically null when the conformal boundary is approached. We show that maximal hypersurfaces which are uniformly spacelike at infinity, so that the last possibility does not occur, are uniquely defined by their asymptotic Dirichlet data, meet the conformal boundary orthogonally, and have a full polyhomogeneous expansion at infinity, in particular a well-defined renormalised volume; see Theorems 6.4 and 8.1 for precise statements.
Our analysis shows that a possible obstruction to regularity of maximal hypersurfaces at the conformal boundary at infinity is that the hypersurface becomes asymptotically null. The key remaining question is to show that asymptotically null maximal hypersurfaces do not exist, or to construct a counterexample.
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