Some aspects of the global geometry of entire space-like submanifolds. (English) Zbl 0998.53036
Let \(M\) be an \(m\)-dimensional closed space-like submanifold in an \((m+n)\)-dimensional pseudo-Euclidean space \(R_n^{m+n}\). The authors prove that if \(M\) has parallel mean curvature and has bounded Gauss image, then \(M\) is a linear subspace.
Reviewer: K.Ogiue (Tokyo)
References:
| [1] | Aiyama, R., The generalized Gauss map of a space-like submanifold with parallel mean curvature vector in pseudo-Euclidean space, Japan J. Math. 20(1) (1994), 93–114. · Zbl 0818.53082 |
| [2] | Calabi, E., Examples of Bernstein problems for some nonlinear equations, Proc. Symp. Global Analysis U.C.Berkeley (1968). · Zbl 0169.53303 |
| [3] | Calabi, E., Improper affine hyperspheres of convex type and generalization of a theorem by K. Jögens’, Michigan Math. J. 5 (1958), 105–126. · Zbl 0113.30104 · doi:10.1307/mmj/1028998055 |
| [4] | Cheng, S. Y. and Yau, S. T., Maximal spacelike hypersurfaces in the Lorentz-Minkowski space, Ann. Math. 104 (1976), 407–419. · Zbl 0352.53021 · doi:10.2307/1970963 |
| [5] | Choi, H. In and Treibergs, A., Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space, J. Differential Geometry 32 (1990), 775–817. · Zbl 0717.53038 |
| [6] | Hitchin, N. J., The moduli space of special Lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci 25(4) (1998), 503–515. · Zbl 1015.32022 |
| [7] | Ishihara, T., The harmonic Gauss maps in a generalized sense, J. London Math. Soc. 26, 104–112 (1982). · Zbl 0498.53041 · doi:10.1112/jlms/s2-26.1.104 |
| [8] | McLean, R. C, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705–747. · Zbl 0929.53027 |
| [9] | Pogorelov, A. V., On the improper convex affine hyperspheres, Geometriae Dedicata 1 (1972), 33–46. · Zbl 0251.53005 · doi:10.1007/BF00147379 |
| [10] | Xin, Y. L., On Gauss image of a spacelike hypersurface with constant mean curvature in Minkowski space, Comment. Math. Helv. 66, (1991), 590–598. · Zbl 0752.53038 · doi:10.1007/BF02566667 |
| [11] | Xin, Y. L., A rigidity theorem for a space-like graph of higher codimension, manuscripta math. 103(2) (2000), 191–202. · Zbl 0989.53026 · doi:10.1007/s002290070020 |
| [12] | Xin, Y. L., Geometry of harmonic maps, Birkhäuser, PNLDE vol. 23, 1996. · Zbl 0848.58014 |
| [13] | Xin, Y. L. and Ye, Rugang, Bernstein-type theorems for space-like surfaces with parallel mean curvature, J. reine angew. Math. 489 (1997), 189–198. · Zbl 0879.53046 |
| [14] | Yau, S. T., Harmonic function on complete Riemannian manifold, Comm. Pure Appl. Math. 28 (1975), 201–228. · Zbl 0291.31002 · doi:10.1002/cpa.3160280203 |
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