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

TL;DR

This paper establishes the existence of horo-convex hypersurfaces in hyperbolic space with prescribed shifted Gauss curvature, using degree theory and a priori estimates, without requiring sign conditions on the curvature functions.

Contribution

It introduces a new existence result for prescribed shifted Gauss curvature equations in hyperbolic space, leveraging horo-convexity and degree theory, differing from prior Weingarten curvature problems.

Findings

Existence of solutions under certain conditions

No sign condition needed for the radial derivative

Extension of curvature prescription methods to hyperbolic space

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 $\mathbb{H}^{n+1}$.