Ginzburg-Landau Equations on Non-compact Riemann Surfaces

TL;DR

This paper investigates solutions to the Ginzburg-Landau equations on non-compact Riemann surfaces with negative curvature, revealing new vortex solutions that break gauge symmetry and generalize superconductivity phenomena.

Contribution

It establishes the existence of novel vortex solutions with energies below the constant curvature case and provides their asymptotic analysis, extending understanding of Ginzburg-Landau equations on curved surfaces.

Findings

Existence of solutions with energy less than the constant curvature magnetic field.

Solutions are non-commutative generalizations of Abrikosov vortex lattices.

Demonstrates spontaneous gauge-translational symmetry breaking.

Abstract

We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.