## Abstract

Two problems concerning maps φ{symbol} with point singularities from a domain Ω C ℝ^{3} to S^{2} are solved. The first is to determine the minimum energy of φ{symbol} when the location and topological degree of the singularities are prescribed. In the second problem Ω is the unit ball and φ{symbol}=g is given on ∂Ω; we show that the only cases in which g(x/|x|) minimizes the energy is g=const or g(x)=±Rx with R a rotation. Extensions of these problems are also solved, e.g. points are replaced by "holes," ℝ^{3}, S^{2} is replaced by ℝ^{N}, S^{N-1} or by ℝ^{N}, ℝP^{N-1}, the latter being appropriate for the theory of liquid crystals.

Original language | English (US) |
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Pages (from-to) | 649-705 |

Number of pages | 57 |

Journal | Communications In Mathematical Physics |

Volume | 107 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1986 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics