Ginzburg-Landau Equations on Non-compact Riemann Surfaces
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.
Keywords
Cite
@article{arxiv.2203.14179,
title = {Ginzburg-Landau Equations on Non-compact Riemann Surfaces},
author = {Nicolas M. Ercolani and Israel Michael Sigal and Jingxuan Zhang},
journal= {arXiv preprint arXiv:2203.14179},
year = {2023}
}
Comments
added Section 7 on extension to degenerate case; reformulated main result; minor changes