A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $

Abstract

<p style='text-indent:20px;'>Inspired by the generalized Christoffel problem, we consider a class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{H}^{n+1} $\end{document}</tex-math></inline-formula>. Under some sufficient conditions, we prove the a priori estimates for solutions to the Monge-Ampère type equation <inline-formula><tex-math id="M3">\begin{document}$ \det(\kappa-\mathbf{1}) = f(X, \nu(X)) $\end{document}</tex-math></inline-formula>. Moreover, we obtain an existence result for the compact horo-convex hypersurface <inline-formula><tex-math id="M4">\begin{document}$ M $\end{document}</tex-math></inline-formula> satisfying the above equation with various assumptions.</p>