Irreducible Ginzburg-Landau fields in dimension 2

Abstract

Ginzburg--Landau fields are the solutions of the Ginzburg--Landau equations which depend on two positive parameters, $\alpha$ and $\beta$. We give conditions on $\alpha$ and $\beta$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in $\rl^2$, spheres, tori, etc.) with de Gennes--Neumann boundary conditions. We also prove that, for each such manifold and all positive $\alpha$ and $\beta$, the Ginzburg--Landau free energy is a Palais--Smale function on the space of gauge equivalence classes, Ginzburg--Landau fields exist for only a finite set of energy values, and the moduli space of Ginzburg--Landau fields is compact.