Smooth solutions to the Christoffel problem in ℍⁿ⁺¹

2024/06/12 by Li Chen, Chen, Li
Mathematics · #Analysis of PDEs (math.AP) #Analytic Number Theory Research #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2406.09449

openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The famous Christoffel problem is possibly the oldest problem of prescribed curvatures for convex hypersurfaces in Euclidean space. Recently, this problem has been naturally formulated in the context of uniformly h-convex hypersurfaces in hyperbolic space by Espinar-Gálvez-Mira. Surprisingly, Espinar-Gálvez-Mira find that the Christoffel problem in hyperbolic space is essentially equivalent to the Nirenberg-Kazdan-Warner problem on prescribing scalar curvature on \mathbbSⁿ. This equivalence opens a new door to study the Nirenberg-Kazdan-Warner problem. In this paper, we establish a existence of solutions to the Christoffel problem in hyperbolic space by proving a full rank theorem. As a corollary, a existence of solutions to the Nirenberg-Kazdan-Warner problem follows.

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