Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Abstract

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in Min3 for datas that are invariant under the action of a co-compact Fuchsian group.-the Minkowski problem in the 3-dimensional Minkowski space.The results concerning the last two items are essentially application of the first one. STATEMENTS OF RESULTS K-slicings of MGHC spacetimes with constant curvature.The following is our main result: Theorem 2.1. Let M be a 3-dimensional non-elementary MGHC spacetime with constant curvature Λ. If Λ ≥ 0, reversing the time orientation if necessary, we assume that M is future complete.• If Λ ≥ 0 (flat case or locally de Sitter case), then M admits a unique K-slicing. The leaves of this slicing are the level sets of a K-time ranging over (−∞, −Λ). • If Λ < 0 (locally anti-de Sitter case), then M does not admit any global K-slicing, but each of the two connected component of the complement of the convex core 2 of M admits a unique K-slicing. The leaves of the K-slicing of the past of the convex core are the level sets of a K-time ranging over (−∞, 0). The leaves of the K-slicing of the future of the convex core are the level sets of a reverse K-time 3 ranging over (−∞, 0).Let us make a few comments on this result.Gaussian curvature. Since the Gaussian curvature of a surface is R = Λ + κ, the Gaussian curvature of the leaves of the K-slicings provided by Theorem 2.1 varies in (−∞, 0) when Λ ≥ 0, and in (−∞, Λ) when Λ < 0.CMC times. It is our interest on CMC-times that led us to extend our attention to more general geometric times. Existence of CMC times on MGHC spacetimes of constant non-positive curvature and any dimension was proved in [3,4]. For spacetimes locally modelled on the de Sitter space, there are some restrictions but not in dimension 3.Regularity. The slicings provided by Theorem 2.1 are continuous. It follows from their uniqueness (i.e. they are canonical). Extra smoothness is not automatic (e.g. the cosmological time is C 1,1 , but not C 2 ).Here, we can hope that our K-slicings are (real) analytic, and even more, they depend analytically on the spacetime (within the space of MGHC spacetimes of curvature Λ and fixed topology). All this depends on consideration of Einstein equations in a K-gauge.Non-standard isometric immersions of H 2 in Min 3 . It was observed by Hano and Nomizu [38], that the hyperbolic plane H 2 admits non standard isometric immersions in the Minkowski space Min 3 (i.e. different from the hyperbola, up to a Lorentz motion).

Hyperbolic ends with particles and grafting on singular surfaces

“…We therefore obtain the following. This gives an affirmative answer to Question 6.4 in [17], and generalizes a result about constant Gauss curvature foliation of future-complete globally hyperbolic maximal compact de Sitter spacetimes (see [4,Theorem 2.1]) to the case with particles.…”

Area-preserving diffeomorphisms of the hyperbolic plane and K-surfaces in anti-de Sitter space

“…Theorem 7.1 [7]. Let φ : ∂H 2 → ∂H 2 be an orientation-preserving homeomorphism which is equivariant for a pair of Fuchsian surface group representation…”

“…Since the groundbreaking work of Mess [30], the interest in the study of anti-de Sitter manifolds has grown, often motivated by the similarities with hyperbolic three-dimensional geometry, and with special emphasis on its relations with Teichmüller theory of hyperbolic surfaces. See for instance [2,6,7,12,15,18,27]. In fact, as outlined in [1,14,37,39], several constructions can be generalized in the context of universal Teichmüller space.…”

Christoffel and Minkowski problems in Minkowski space

“…The case d = 2 was obtained in [2] by totally different methods than in [7]. They use dimensional specificities of the dimension 2 from the point of view of geometry and topology of surfaces.…”