Classification of solutions to the isotropic horospherical $p$-Minkowski problem in hyperbolic plane

Abstract

In \cite{LX}, the first author and Xu introduced and studied the horospherical $p$-Minkowski problem in hyperbolic space ${\mathbb{H}}^{n+1}$. In particular, they established the uniqueness result for solutions to this problem when the prescribed function is constant and $p\ge -n$. This paper focuses on the isotropic horospherical $p$-Minkowski problem in hyperbolic plane ${\mathbb{H}}^2$, which corresponds to the equation \begin{equation}\label{0} φ^{-p}\left(φ_{θθ}-\frac{φ_θ^2}{2φ}+\frac{φ-φ^{-1}}{2}\right)=γ\quad\text{on}\ \mathbb{S}^1, \end{equation} where ${\hat{I}}^3$ is a positive constant. We provide a classification of solutions to the above equation for $p\ge -7$, as well as a nonuniqueness result of solutions for $p<-7$. Furthermore, we extend this problem to the isotropic horospherical $q$-weighted $p$-Minkowski problem in hyperbolic plane and derive some uniqueness and nonuniqueness results.

19 pages, 2 figures. All comments are welcome