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 Hn+1. This problem generalizes the classical Lp Christoffel-Minkowski problem in Euclidean space Rn+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 Hn+1 reduces to a Nirenberg-type problem on Sn in conformal geometry. As a consequence, our result also provides the existence of solutions to this Nirenberg-type problem.