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

TL;DR

This paper classifies solutions to the isotropic horospherical p-Minkowski problem in hyperbolic plane for various p values, revealing uniqueness for p ≥ -7 and nonuniqueness for p < -7, and extends to a weighted version.

Contribution

It provides a complete classification of solutions for the isotropic horospherical p-Minkowski problem in hyperbolic plane, including new nonuniqueness results for p < -7.

Findings

Classified solutions for p ≥ -7

Established nonuniqueness for p < -7

Extended to a weighted p-Minkowski problem

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} \varphi^{-p}\left(\varphi_{\theta\theta}-\frac{\varphi_{\theta}^2}{2\varphi}+\frac{\varphi-\varphi^{-1}}{2}\right)=\gamma\quad\text{on}\ \mathbb{S}^1, \end{equation} where $\gamma$ 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.