Singularities of energy minimizing maps from the ball to the sphere: Examples, counterexamples, and bounds. (English) Zbl 0673.58013

Almgren, Frederick J. jun.; Lieb, Elliot H.

Ann. Math. (2) 128, No. 3, 483-530 (1988).

The paper studies the singularities of Dirichlet’s integral energy upon the mappings from a domain in \(R^ 3\) to \(S^ 2\) with prescribed boundary value functions. The main result in the first part of the paper states that the number of singular points of such a minimizer is linearly dominated by the boundary energy. The second part of the paper is devoted to “the curious behaviour of singularities”. It is shown that, within an open ball in \(R^ 3\), the minimizers with zero boundary mapping area can have arbitrarily many singularities. It is then constructed a boundary function having a symmetry about the equator in a ball for which every minimizer possesses unsymmetric singularities. Many other interesting results are proved.

Reviewer: D.Motreanu

MSC:

58E20	Harmonic maps, etc.
58C99	Calculus on manifolds; nonlinear operators
82D30	Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
28A99	Classical measure theory

Keywords:

minimizing map; harmonic map; liquid crystals; Dirichlet’s integral energy; singular points; unsymmetric singularities