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

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

2006-12-11

arxiv.org/abs/math/0612286v2

Mathematics

Differential Geometry

18 pages

Scientific paper

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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