# Ginzburg-Landau equations on Riemann surfaces of higher genus

## TL;DR

This paper investigates the Ginzburg-Landau equations on higher genus Riemann surfaces, explicitly constructing solutions near constant curvature states, classifying related holomorphic structures, and analyzing energy properties.

## Contribution

It provides explicit local moduli space constructions, a classification of holomorphic structures, and energy analysis for solutions on higher genus Riemann surfaces.

## Key findings

- Explicit construction of local solutions near constant curvature states
- Classification of holomorphic structures via degree, Abel-Jacobi map, and symmetric products
- Identification of energy bounds relative to constant curvature solutions

## 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 a neighbourhood 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.