The Minkowski problem, new constant curvature surfaces in R³, and some applications

2012/04/20 by Antonio Alarcón, Antonio Alarcon, Alarcon, Antonio +2
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.DG

paper · pdf · doi:10.48550/arxiv.1204.4687

17 pages, 1 figure

openalex publication_date 2012/04/20 · arxiv created 2013/02/17 · arxiv updated 2013/02/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let m∈ℕ, m≥ 2, and let \p_j\_j=1^m be a finite subset of \mathbbS² such that 0∈ℝ³ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies K in ℝ³ whose boundary surface consists of an open surface S with constant Gauss curvature (respectively, constant mean curvature) and m planar compact discs D₁,...,D_m, such that the Gauss map of S is a homeomorphism onto \mathbbS²-\p_j\_j=1^m and D_j\bot p_j, for all j. We derive applications to the generalized Minkowski problem, existence of harmonic diffeomorphisms between domains of \mathbbS², existence of capillary surfaces in ℝ³, and a Hessian equation of Monge-Ampere type.

Citations

• Surfaces of constant curvature in R³ with isolated singularities

Related

• Surfaces of prescribed linear Weingarten curvature in ℝ³
• Geometric properties of surfaces with the same mean curvature in R³ and L³
• Curvature bound for L_p Minkowski problem
• Existence of free boundary disks with constant mean curvature in ℝ³
• Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map