Spherical Surfaces

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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.
Original languageEnglish
JournalExperimental Mathematics
Volume25
Issue number3
Pages (from-to)257-272
ISSN1058-6458
DOIs
Publication statusPublished - 2016

Keywords

  • Differential geometry
  • Integrable systems
  • Loop groups
  • Spherical surfaces
  • Constant Gauss curvature
  • Singularities
  • Cauchy problem

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