Asymptotic behavior of Cauchy hypersurfaces in constant curvature space-times. (English) Zbl 1380.53080
Summary: We study the asymptotic behavior of convex Cauchy hypersurfaces on maximal globally hyperbolic spatially compact space-times of constant curvature. We generalise the result of the author [Ann. Inst. Fourier 64, No. 2, 457–466 (2014; Zbl 1336.53033)] to the \((2+1)\) de Sitter and anti de Sitter cases. We prove that in these cases the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a real tree. Moreover, this limit does not depend on the choice of the time function. We also consider the problem of asymptotic behavior in the flat \((n+1)\) dimensional case. We prove that the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a \(\mathrm{CAT}(0)\) metric space. Moreover, this limit does not depend on the choice of the time function.
MSC:
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 53C40 | Global submanifolds |
Keywords:
Lorentzian geometry; constant curvature spacetime; quasi-concave time function; equivariant Gromov topologyCitations:
Zbl 1336.53033References:
| [1] | Andersson, L.: Constant mean curvature foliations of flat space-times. Comm. Anal. Geom. 10(5), 1125-1150 (2002) · Zbl 1038.53025 · doi:10.4310/CAG.2002.v10.n5.a10 |
| [2] | Andersson, L.: Constant mean curvature foliations of simplicial flat spacetimes. Comm. Anal. Geom. 13(5), 963-979 (2005) · Zbl 1123.53034 · doi:10.4310/CAG.2005.v13.n5.a6 |
| [3] | Andersson, L., Barbot, T., Béguin, F., Zeghib, A.: Cosmological time versus CMC time in spacetimes of constant curvature. Asian J. Math. 1, 37-88 (2012) · Zbl 1246.53093 · doi:10.4310/AJM.2012.v16.n1.a2 |
| [4] | Andersson, L., Galloway, G.J., Howard, R.: The cosmological time function. Class. Quantum Gravity 15(2), 309-322 (1998) · Zbl 0911.53039 · doi:10.1088/0264-9381/15/2/006 |
| [5] | Andersson, L., Moncrief, V., Tromba, A.J.: On the global evolution problem in \[2+12+1\] gravity. J. Geom. Phys. 23(3-4), 191-205 (1997) · Zbl 0898.58003 · doi:10.1016/S0393-0440(97)87804-7 |
| [6] | Barbot, T.: Globally hyperbolic flat space-times. J. Geom. Phys. 53(2), 123-165 (2005) · Zbl 1087.53065 · doi:10.1016/j.geomphys.2004.05.002 |
| [7] | Barbot, T., Béguin, F., Zeghib, A.: Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes, application to the Minkowski problem in the Minkowski space. Ann. Inst. Fourier 61(2), 511-591 (2011) · Zbl 1234.53019 · doi:10.5802/aif.2622 |
| [8] | Barbot, T., Béguin, F., Zeghib, A.: Feuilletages des espaces temps globalement hyperboliques par des hypersurfaces à courbure moyenne constante. C. R. Math. Acad. Sci. Paris 336(3), 245-250 (2003) · Zbl 1026.53015 · doi:10.1016/S1631-073X(03)00019-0 |
| [9] | Barbot, T., Béguin, F., Zeghib, A.: Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on \[{\rm AdS}_3\] AdS3. Geom. Dedicata 126, 71-129 (2007) · Zbl 1255.83118 · doi:10.1007/s10711-005-6560-7 |
| [10] | Belraouti, M.: Convergence asymptotique des niveaux de temps quasi-concave dans un espace temps à courbure constante. Ph.D. thesis, (2013) · Zbl 1087.53065 |
| [11] | Belraouti, M.: Sur la géométrie de la singularité initiale des espaces-temps plats globalement hyperboliques. Annales de l’institut Fourier 64(2), 457-466 (2015) · Zbl 1336.53033 · doi:10.5802/aif.2854 |
| [12] | Benedetti, R., Bonsante, F.: Canonical Wick rotations in 3-dimensional gravity. Mem. Amer. Math. Soc., 198(926), viii+164, (2009) · Zbl 1165.53047 |
| [13] | Benedetti, R., Guadagnini, E.: Cosmological time in \[(2+1)(2+1)\]-gravity. Nuclear Phys. B 613(1-2), 330-352 (2001) · Zbl 0970.83039 · doi:10.1016/S0550-3213(01)00386-8 |
| [14] | Bonsante, F.: Flat spacetimes with compact hyperbolic Cauchy surfaces. J. Differ. Geom. 69(3), 441-521 (2005) · Zbl 1094.53063 · doi:10.4310/jdg/1122493997 |
| [15] | Bonsante, F., Fillastre, F.: The equivariant Minkowski problem in Minkowski space. arxiv:1405.4376 · Zbl 1421.52002 |
| [16] | Bonsante, F., Seppi, A.: On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry. arxiv:1501.04922 · Zbl 1336.53034 |
| [17] | Bridson, M.R., Haefliger, A.: Metric Spaces of Non-Positive Curvature, Volume 319 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1999) · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9 |
| [18] | Geroch, R., Kronheimer, E.H., Penrose, R.: Ideal points in space-time. Proc. R. Soc. Lond. Ser. A 327, 545-567 (1972) · Zbl 0257.53059 · doi:10.1098/rspa.1972.0062 |
| [19] | Geroch, R.: Domain of dependence. J. Math. Phys. 11, 437-449 (1970) · Zbl 0189.27602 · doi:10.1063/1.1665157 |
| [20] | Hawking, S.W.: The occurrence of singularities in cosmology. I. Proc. R. Soc. Ser. A 294, 511-521 (1966) · Zbl 0139.45803 · doi:10.1098/rspa.1966.0221 |
| [21] | Hawking, S.W.: The occurrence of singularities in cosmology. II. Proc. R. Soc. Ser. A 295, 490-493 (1966) · Zbl 0148.46504 · doi:10.1098/rspa.1966.0255 |
| [22] | Hawking, S.W., Ellis, G.F.R., The large scale structure of space-time. Cambridge University Press, London, : Cambridge Monographs on Mathematical. Physics, No 1. (1973) · Zbl 0265.53054 |
| [23] | Hawking, S.W., Penrose, R.: The singularities of gravitational collapse and cosmology. Proc. R. Soc. Lond. Ser. A 314, 529-548 (1970) · Zbl 0954.83012 · doi:10.1098/rspa.1970.0021 |
| [24] | Mess, G.: Lorentz spacetimes of constant curvature. Geom. Dedicata 126, 3-45 (2007) · Zbl 1206.83117 · doi:10.1007/s10711-007-9155-7 |
| [25] | Moncrief, V.: Reduction of the Einstein equations in \[2+12+1\] dimensions to a Hamiltonian system over Teichmüller space. J. Math. Phys. 30(12), 2907-2914 (1989) · Zbl 0704.53076 · doi:10.1063/1.528475 |
| [26] | Morgan, J.W., Shalen, P.B.: Valuations, trees, and degenerations of hyperbolic structures. I. Ann. Math. (2) 120(3), 401-476 (1984) · Zbl 0583.57005 · doi:10.2307/1971082 |
| [27] | O’Neill, B.: Semi-Riemannian Geometry: With Applications to Relativity. Academic Press, London (1983) · Zbl 0531.53051 |
| [28] | Otal, J. P.: Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3. Astérisque, (235):x+159, (1996) · Zbl 0855.57003 |
| [29] | Paulin, F.: Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94(1), 53-80 (1988) · Zbl 0673.57034 · doi:10.1007/BF01394344 |
| [30] | Paulin, F.: The Gromov topology on \[{ R}\] R-trees. Topol. Appl. 32(3), 197-221 (1989) · Zbl 0675.20033 · doi:10.1016/0166-8641(89)90029-1 |
| [31] | Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57-59 (1965) · Zbl 0125.21206 · doi:10.1103/PhysRevLett.14.57 |
| [32] | Scannell, K.P.: Flat conformal structures and the classification of de Sitter manifolds. Comm. Anal. Geom. 7(2), 325-345 (1999) · Zbl 0941.53040 · doi:10.4310/CAG.1999.v7.n2.a6 |
| [33] | Schmidt, B.G.: A new definition of singular points in general relativity. Gen. Relat. Gravit, 1(3):269-280, (1970/71) · Zbl 0332.53039 |
| [34] | Treibergs, A.E.: Entire spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent. Math. 66, 39-56 (1982) · Zbl 0483.53055 · doi:10.1007/BF01404755 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
