Spherical Surfaces

TL;DR

This paper investigates surfaces of constant positive Gauss curvature in Euclidean 3-space using harmonic maps and loop group techniques, providing new solutions, singularity criteria, and symmetry classifications.

Contribution

It introduces a loop group framework to solve geometric Cauchy problems for spherical surfaces, including singularity analysis and symmetry considerations.

Findings

Derived criteria for generic singularities on spherical surfaces

Computed explicit examples of spherical surfaces with singularities

Identified loop group potentials for symmetric and singular surfaces

Abstract

We study 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.