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Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space. (English) Zbl 1097.53040
The paper deals with space-like convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space \(\mathbb R_1^n\) where \(n \geq 2.\) More explicitly, the authors study the entire solutions on \(\mathbb R^n\) of the Monge-Ampère equation: \[ \text{ det} D^2 u = \psi(x,u)(1-| Du| ^2)^{\frac{n+2}{2}} \] with the space-like condition \[ | Du| <1, \] because each of these hypersurfaces can be expressed as the graph of a convex function \(x_{n+1}=u(x)\) satisfying the Monge-Ampère equation and the space-like condition where \(\psi\) is the Gauss curvature.
The authors provide a classification for entire space-like K-hypersurfaces admitting a rotational symmetry with respect to a space-like axis. They are also interested in determining entire K-hypersurfaces in Minkowski space with bounded Gauss curvature admitting a prescribed tangent cone at infinity.
Moreover, the Minkowski type problem is considered due to its close relation to the problem of finding entire space-like hypersurfaces with prescribed Gauss curvature and extensions of some of the results in [A.-M. Li, Arch. Math. 64, No. 6, 534–551 (1995; Zbl 0828.53050)] are obtained.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
35J60 Nonlinear elliptic equations
49Q10 Optimization of shapes other than minimal surfaces
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI arXiv
[1] DOI: 10.1007/BF01211061 · Zbl 0512.53055 · doi:10.1007/BF01211061
[2] DOI: 10.2307/1971509 · Zbl 0704.35045 · doi:10.2307/1971509
[3] DOI: 10.2307/1971510 · Zbl 0704.35044 · doi:10.2307/1971510
[4] DOI: 10.1002/cpa.3160370306 · Zbl 0598.35047 · doi:10.1002/cpa.3160370306
[5] DOI: 10.2307/1970963 · Zbl 0352.53021 · doi:10.2307/1970963
[6] Choi H. I., J. Di\currency. Geom. 32 pp 775– (1990)
[7] DOI: 10.1007/BF01060827 · Zbl 0724.35039 · doi:10.1007/BF01060827
[8] DOI: 10.1090/S0002-9947-98-02079-0 · Zbl 0919.35046 · doi:10.1090/S0002-9947-98-02079-0
[9] DOI: 10.1002/cpa.20010 · Zbl 1066.53109 · doi:10.1002/cpa.20010
[10] DOI: 10.1007/BF01455315 · Zbl 0507.53042 · doi:10.1007/BF01455315
[11] DOI: 10.1007/BF01195136 · Zbl 0828.53050 · doi:10.1007/BF01195136
[12] DOI: 10.1007/BF01404755 · Zbl 0483.53055 · doi:10.1007/BF01404755
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