Quantization and non-quantization of energy for higher-dimensional Ginzburg-Landau vortices

Given a family of critical points for the complex Ginzburg-Landau energies on a manifold M, with natural energy growth, it is known that the vorticity sets converge subsequentially to the support of a stationary, rectifiable varifold in the interior, characterized as the concentrated portion of the limit of the normalized energy measures. When the dimension is two or the solutions are energy-minimizing, it is known moreover that this varifold is integral. In the present paper, we show that for a general family of critical points in dimension at least three, this energy quantization phenomenon only holds where the density is less than 2, and show that this is sharp, in the sense that for any assigned real density higher than 2, there exists a family of critical points in the ball with concentration varifold given by a plane with this density.