A geometric Ginzburg-Landau problem

In contrast, the functional studied here involves the function Ψ(ν) = ν2 1 , which attains its minimum on the whole equator {0} × S1. At a first glance, this problem looks closer to the theory of Ginzburg-Landau vortices, which was first developed by Bethuel, Brezis, and H ́elein [2]. But despite the formal similarities, we will see that the analogy between these two problems is limited, too. Above all, the interesting phenomena occur at different scaling regimes for the two energies. Ginzburg-Landau vortices have a typical energy of order | log |. For the geometric problem, unless we allow the surfaces to shrink to a point in the limit, we need at least an energy of order −1/2.