The horospherical p-Christoffel–Minkowski problem in hyperbolic space

The horospherical $p$-Christoffel–Minkowski problem, introduced by Li and Xu (2022), involves prescribing the $k$-th horospherical $p$-surface area measure for $h$-convex domains in hyperbolic space $H^{n+1}$. This problem generalizes the classical $L^p$ Christoffel–Minkowski problem in Euclidean space $R^{n+1}$. In this paper, we study a fully nonlinear equation associated with this problem and establish the existence of a uniformly $h$-convex solution under suitable assumptions on the prescribed function. The proof relies on a full rank theorem, which we demonstrate using a viscosity approach inspired by the work of Bryan et al. (2023).

When $p=0$, the horospherical $p$-Christoffel–Minkowski problem in $H^{n+1}$ reduces to a Nirenberg-type problem on $S^n$ in conformal geometry. As a consequence, our result also provides the existence of solutions to this Nirenberg-type problem.