Entire spacelike hypersurfaces with constant σ_k curvature in Minkowski space

2020/07/03 by Wang, Zhizhang, Xiao, Ling
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.01495

Abstract

In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant σ_k curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant σ_k curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn't exist any entire, convex, strictly spacelike, constant σ_k curvature hypersurfaces. Moreover, we generalize the result in \citeRWX and construct strictly convex, spacelike, constant σ_k curvature hypersurface with bounded principal curvature, whose image of the Gauss map is the unit ball.

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