Surfaces of constant curvature in R^3 with isolated singularities

Details Surfaces of constant curvature in R^3 with isolated singularities Surfaces of constant curvature in R^3 with isolated singularities

2010-07-15

arxiv.org/abs/1007.2523v1

Mathematics

Differential Geometry

28 pages

Scientific paper

We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 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 R^3, 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.

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