The energy of Hopf vector fields

C M Wood

The energy of Hopf vector fields

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields.

Original language	English
Pages (from-to)	71-88
Number of pages	18
Journal	Manuscripta Mathematica
Volume	101
Issue number	1
Publication status	Published - Jan 2000

Keywords

HARMONIC MAPPINGS
SPHERES
VOLUMES
MAPS

Cite this

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AB - The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields.

KW - HARMONIC MAPPINGS

KW - SPHERES

KW - VOLUMES

KW - MAPS

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VL - 101

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JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

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