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 language | English |
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Pages (from-to) | 101-113 |
Number of pages | 13 |
Journal | Annals of Global Analysis and Geometry |
Volume | 34 |
Issue number | 2 |
DOIs | |
Publication status | Published - 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