Critical points of the Ginzburg–Landau system on a Riemannian surface

Abstract

We study here the Ginzburg-Landau functional on a Riemannian surface S endowed with an arbitrary metric

E_ε(u)=½∫_𝒮||∇u||²+1/(4ε²)∫_𝒮(|u|²−1)², u∈H¹(S,C),

where ε∈]0,+∞[. We study the asymptotic behaviour of the critical points for E_ε as ε→0. We define a renormalized energy which allows to characterize the position of the singularities at the limit.