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

2024/05/07 by Haizhong Li, Yao Wan, Li, Haizhong +1
Mathematics · #34C25 #52A55 #53A04 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2405.04301

openalex publication_date 2024/05/07 · openalex created_date 2024/05/11 · openalex updated_date 2026/07/28

Abstract

In \citeLX, the first author and Xu introduced and studied the horospherical p-Minkowski problem in hyperbolic space ℍⁿ⁺¹. In particular, they established the uniqueness result for solutions to this problem when the prescribed function is constant and p≥ -n. This paper focuses on the isotropic horospherical p-Minkowski problem in hyperbolic plane ℍ², which corresponds to the equation φ^-p(φ_θθ-(φ_θ²)/(2φ)+\fracφ-φ^-12)=γ\quadon \mathbbS¹, where γ is a positive constant. We provide a classification of solutions to the above equation for p≥ -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.

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