Vortices for the magnetic Ginzburg-Landau theory in curved space

TL;DR

This paper investigates vortex solutions in the magnetic Ginzburg-Landau theory within curved space, establishing existence theorems and analyzing properties of solutions influenced by gravity and curvature effects.

Contribution

It introduces new existence theorems for vortex solutions in curved space, employing novel methods to handle singularities and gravitational effects in the Ginzburg-Landau framework.

Findings

Existence of vortex solutions with quantized flux and total curvature linked to vortex number.

Development of a constraint minimization and monotone iteration method for singular PDEs.

Construction of radially symmetric vortex solutions using shooting and fixed-point methods.

Abstract

Since the Ginzburg-Landau theory is concerned with macroscopic phenomena, and gravity affects how objects interact at the macroscopic level. It becomes relevant to study the Ginzburg-Landau theory in curved space, that is, in the presence of gravity. In this paper, some existence theorems are established for the vortex solutions of the magnetic Ginzburg-Landau theory coupled to the Einstein equations. First, when the coupling constant \lambda=1, we get a self-dual structure from the Ginzburg-Landau theory, then a partial differential equation with a gravitational term that has power-type singularities is deduced from the coupled system. To overcome the difficulty arising from the orders of singularities at the vortices, a constraint minimization method and a monotone iteration method are employed. We also show that the quantized flux and total curvature are determined by the number of vortices. Second, when the coupling constant \lambda>0, we use a suitable ansatz to get the radially symmetric case for the magnetic Ginzburg-Landau theory in curved space. The existence of the symmetric vortex solutions are obtained through combining a two-step iterative shooting argument and a fixed-point theorem approach. Some fundamental properties of the solutions are established via applying a series of analysis techniques.