Bending and stretching unit vector fields in Euclidean and hyperbolic 3-space

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Abstract

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harmonic maps from 3-dimensional Euclidean domains to the 2-sphere. Finally, the harmonic unit vector field tangent to a parallel family of hyperbolic geodesics is shown to be unstable, by constructing a class of compactly supported energy-decreasing variations. All examples considered have infinite total bending.

Original languageEnglish
Pages (from-to)101-113
Number of pages13
JournalAnnals of Global Analysis and Geometry
Volume34
Issue number2
DOIs
Publication statusPublished - Sept 2008

Keywords

  • harmonic section
  • harmonic map
  • harmonic unit vector field
  • harmonic function
  • sine-gordon equation
  • pendulum equation
  • horospheres
  • energy
  • bending
  • unstable
  • stretchable
  • RIEMANNIAN-MANIFOLDS
  • HARMONIC MAPPINGS
  • ENERGY
  • VOLUME

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