Harmonic maps with defects

Two problems concerning maps φ{symbol} with point singularities from a domain Ω C ℝ³ to S² 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," ℝ³, S² 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.