Surfaces of constant curvature in R³ with isolated singularities

2010/07/15 by José A. Gálvez, Jose A. Galvez, Laurent Hauswirth +4 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1007.2523

28 pages

arxiv created 2010/07/15 · openalex publication_date 2010/07/15 · arxiv updated 2010/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that finite area isolated singularities of surfaces with constant positive curvature in R³ are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves in the 2-sphere with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R³, i.e. compact convex surfaces of constant positive curvature with a finite number of singularities, and give applications to harmonic maps and constant mean curvature surfaces.

Cited by

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

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