Horo-convex hypersurfaces with prescribed shifted Gauss curvatures in ${\mathbb{H}}^{n+1}$

Abstract

In this paper, we consider prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in ${\mathbb{H}}^{n+1}$. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the equations. Different from the prescribed Weingarten curvature problem in space forms, we do not impose a sign condition for radial derivative of the functions in the right-hand side of the equations to prove the existence due to the horo-covexity of hypersurfaces in ${\mathbb{H}}^{n+1}$.

arXiv admin note: text overlap with arXiv:1910.10903, arXiv:2004.12066