Spherical Surfaces

Abstract

We study the surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the geometric Cauchy data for the generic singularities, as well as for the cuspidal beaks and cuspidal butterfly singularities. We consider the bifurcations of generic one-parameter families of spherical fronts and provide evidence that suggests that these are the cuspidal beaks, cuspidal butterfly, and one other singularity. We also give the loop group potentials for spherical surfaces with finite-order rotational symmetries and for surfaces with embedded isolated singularities.

KEYWORDS:

• differential geometry
• integrable systems
• loop groups
• spherical surfaces
• constant Gauss curvature
• singularities
• Cauchy problem

2000 AMS Subject Classification:

• Primary 53A05
• 53C43
• Secondary 53C42
• 57R45