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Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space. (English) Zbl 1114.35069
The author proves existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a light-cone. His approach of the existence such hpersurfaces relies in a crucial way on previous works concerning the Dirichlet problem [P. Bayard, Calc. Var. Partial Differ. Equ. 18, No. 1, 1–30 (2003; Zbl 1043.53027); J. Urbas, Calc. Var. Partial Differ. Equ. 18, No. 3, 307–316 (2003; Zbl 1080.53062)].

35J60 Nonlinear elliptic equations
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI arXiv
[1] Bartnik, R., Existence of maximal surfaces in asymptotically flat spacetimes, Commun. Math. Phys., 94, 155-175, (1984) · Zbl 0548.53054
[2] Bartnik, R., Maximal surfaces and general relativity, geometry and partial differential equations, 2nd miniconf., Canberra/aust. 1986, Proc. Cent. Math. Anal. Aust. Natl. Univ., 12, 24-49, (1987)
[3] Bayard, No article title, Calc. Var., 18, 1, (2003) · Zbl 1043.53027
[4] Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 2. Wiley Interscience Publishers (1962) · Zbl 0099.29504
[5] Cheng, S.Y.; Yau, S.T., Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math., 104, 407-419, (1976) · Zbl 0352.53021
[6] Delanoë, Ph., Dirichlet problem for the equation of prescribed Lorentz-Gauss curvature, Ukr. Math. J., 42, 1538-1545, (1990) · Zbl 0724.35039
[7] Delanoë, Ph.: Personal communication. Arch (2005) · Zbl 0919.35046
[8] Guan, B., The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Trans. Amer. Math. Soc., 350, 4955-4971, (1998) · Zbl 0919.35046
[9] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, second edition, Revised Third Printing. Springer-Verlag (1998) · Zbl 0352.53021
[10] Li, A.M., Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space, Arch. Math., 64, 534-551, (1995) · Zbl 0828.53050
[11] Lin, M.; Trudinger, N.S., On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc., 50, 317-326, (1994) · Zbl 0855.26006
[12] Potter, A.J.B., An elementary version of the Leray-Schauder theorem, J. London Math. Soc., 5, 414-416, (1972) · Zbl 0242.47037
[13] Schnürer, O., The Dirichlet problem for Weingarten hypersurfaces in Lorentzian manifolds, Math. Z., 242, 159-181, (2002) · Zbl 1042.53026
[14] Treibergs, A., Entire spacelike hypersurfaces of constant Mean curvature in Minkowski space, Invent. Math., 66, 39-56, (1982) · Zbl 0483.53055
[15] Urbas, J., The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space, Calc. Var., 18, 307-316, (2003) · Zbl 1080.53062
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