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.
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Wood, C.M. Bending and stretching unit vector fields in Euclidean and hyperbolic 3-space. Ann Glob Anal Geom 34, 101–113 (2008). https://doi.org/10.1007/s10455-008-9102-3
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DOI: https://doi.org/10.1007/s10455-008-9102-3
Keywords
- Harmonic section
- Harmonic map
- Harmonic unit vector field
- Harmonic function
- Sine-Gordon equation
- Pendulum equation
- Horospheres
- Energy
- Bending
- Unstable
- Stretchable