Symmetry of the Ginzburg Landau Minimizer in a Disc

The Ginzburg-Landau energy minimization problem for a vector field on a two dimensional disc is analyzed. This is the simplest nontrivial example of a vector field minimization problem and the goal is to show that the energy minimizer has the full geometric symmetry of the problem. The standard methods that are useful for similar problems involving real valued functions cannot be applied to this situation. Our main result is that the minimizer in the class of symmetric fields is stable, i.e., the eigenvalues of the second variation operator are all nonnegative.