Ginzburg–Landau equations on Riemann surfaces of higher genus

Abstract

We study the Ginzburg–Landau equations on Riemann surfaces of arbitrary genus. In particular, we–construct explicitly the (local moduli space of gauge-equivalent) solutions in the neighborhood of the constant curvature ones;–classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel–Jacobi map, and symmetric products of the surface;–determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions.