Constant Mean Curvature Foliations of Globally Hyperbolic Spacetimes Locally Modelled on AdS 3

Abstract

We prove that every three-dimensional maximal globally hyperbolic spacetime, locally modelled on the anti-de Sitter space AdS 3 , with closed orientable Cauchy surfaces, admits a unique CMC time function. (2000). 51M10. Mathematics Subject Classifications

Constant mean curvature foliation of domains of dependence in 𝐴𝑑𝑆₃

“…Namely, we construct this H-surface S H as a limit of H-surfaces (S H ) n , with asymptotic boundary Γ n , with the property that Γ n is the graph of a quasi-symmetric homeomorphism conjugating two cocompact Fuchsian groups and Γ n converges to Γ in the Hausdorff topology. The existence of this approximating sequence (S H ) n is a consequence of some results in [BBZ07] and [BS16].…”

“…Theorem 3.2 (Thm 1.1 in [BBZ07]). Let Γ be a quasi-circle which is the graph of a quasi-symmetric homeomorphism that conjugates two cocompact Fuchsian groups.…”

“…Lemma 5.1 (Maximum Principle, Lemma 2.3 in [BBZ07]). Let Σ and Σ ′ be smooth space-like surfaces in a time-oriented Lorentzian manifold M .…”

“…In fact, as a consequence of the work [BBZ07], the existence (and uniqueness) of a constant mean curvature foliation is known for a particular class of quasi-circles:…”

Maximal surfaces in anti-de Sitter 3-manifolds with particles

“…In [4], the authors proved the existence of a unique maximal space-like surface (that is an area-maximizing surface whose induced metric is Riemannian) in each AdS GHM metric on Σ × R. Note that maximal surfaces are the Lorentzian analogue of minimal surfaces in Riemannian geometry: they are characterized by the vanishing of the mean curvature field. This TOME 66 (2016), FASCICULE 4 result is actually equivalent to the result of R. Schoen of existence of a unique minimal Lagrangian diffeomorphism (see [1]).…”