Classification of solutions to the isotropic horospherical $p$-Minkowski problem in hyperbolic plane
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866929337147588608 |
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| author | Li, Haizhong Wan, Yao |
| author_facet | Li, Haizhong Wan, Yao |
| contents | 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 $γ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04301 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of solutions to the isotropic horospherical $p$-Minkowski problem in hyperbolic plane Li, Haizhong Wan, Yao Differential Geometry Analysis of PDEs Metric Geometry 52A55, 53A04, 34C25 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 $γ$ 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. |
| title | Classification of solutions to the isotropic horospherical $p$-Minkowski problem in hyperbolic plane |
| topic | Differential Geometry Analysis of PDEs Metric Geometry 52A55, 53A04, 34C25 |
| url | https://arxiv.org/abs/2405.04301 |