Abstract
In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant \(\sigma _k\) curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant \(\sigma _k\) curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn’t exist any entire, convex, strictly spacelike, constant \(\sigma _k\) curvature hypersurfaces. Moreover, we generalize the result in Ren et al. (Entire spacelike hypersufaces with constant \(\sigma _{n-1}\) curvature in Minkowski space. Preprint, arXiv:2005.06109) and construct strictly convex, spacelike, constant \(\sigma _k\) curvature hypersurface with bounded principal curvature, whose image of the Gauss map is the unit ball.
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Communicated by F. C. Marques.
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Research of Z. Wang is sponsored by Natural Science Foundation of Shanghai, no. 20JC1412400, 20ZR1406600; supported by NSFC Grants no. 11871161, 11771103; and partially sponsored by Shanghai Rising-Star Program 19QA1400900.
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Wang, Z., Xiao, L. Entire spacelike hypersurfaces with constant \(\sigma _k\) curvature in Minkowski space. Math. Ann. 382, 1279–1322 (2022). https://doi.org/10.1007/s00208-021-02317-0
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DOI: https://doi.org/10.1007/s00208-021-02317-0