zbMATH — the first resource for mathematics

Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space. (English) Zbl 0828.53050
It is well known that there is a standard embedding of the hyperbolic space \(H^2 (-1)\) into the Minkowski space \(\mathbb{R}^3_1\) as the spacelike surface \[ x^2_1 + x^2_2 - x^2_3 = -1\quad (x_3 > 0). \] Denoting the Gauss-Kronecker curvature with \(K\), the intrinsic Gauss-curvature (with respect to the Minkowski inner product) is \(-K\). The present paper deals with the problem of classifying the complete, spacelike, convex hypersurfaces with constant Gauss-Kronecker curvature \(K = 1\); the author restricts the discussion to the case of bounded principal curvatures. The main theorem says that, on the one hand, for every hypersurface in \(\mathbb{R}^{n + 1}_1\) of the above type there exists a function \(\varphi \in C^\infty (\partial B^n(1))\) \((B^n(1)\) the unit Ball), solving a certain Monge-Ampere equation, and conversely, given such a \(\varphi \in C^\infty\) \((\partial B^n (1))\) one can construct a hypersurface of the above type.
Reviewer: F.Manhart (Wien)

53C40 Global submanifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text: DOI
[1] L. Caffarelli, L. Nirenberg andJ. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, 1., Monge-Ampere equation. Comm. Pure Appl. Math.37, 369-402 (1984). · Zbl 0598.35047 · doi:10.1002/cpa.3160370306
[2] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. J√∂rgens. Michigan Math. J.5, 105-126 (1958). · Zbl 0113.30104 · doi:10.1307/mmj/1028998055
[3] S. Y. Cheng andS. T. Yau, On the regularity of the solution of then-dimensional Minkowski problem. Comm. Pure Appl. Math.19, 495-516 (1976). · Zbl 0363.53030 · doi:10.1002/cpa.3160290504
[4] S. Y. Cheng andS. T. Yau, On the regularity of the Monge-Ampere equation det \(\left( {\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}} \right) = F(x,u)\) . Comm. Pure Appl. Math.30, 41-68 (1977). · Zbl 0347.35019 · doi:10.1002/cpa.3160300104
[5] H. I. Choi andA. Treibergs, Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space. J. Differential Geom.32, 775-817 (1990). · Zbl 0717.53038
[6] J.-I. Hano andK. Nomizu, On isometric immersions of the hyperbolic plane into the Lorentz-Minkowski space and the Monge-Ampere equation of a certain type. Math. Ann.262, 245-253 (1983). · Zbl 0507.53042 · doi:10.1007/BF01455315
[7] V. I. Oliker andU. Simon, Codazzi tensor and equations of Monge-Ampere type on compact manifolds of constant sectional curvature. J. Reine Angew. Math.,342, 35-64 (1983). · Zbl 0502.53038
[8] A. V.Pogorelov, The Minkowski multidimensional problem. (Russian) Moscow 1975; Engl. transl.: New York 1978. · Zbl 0355.60064
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.