Vortices for the magnetic Ginzburg–Landau theory in curved space

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 λ = 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 minimisation method and a monotone iteration method are employed. We also show that the quantised flux and total curvature are determined by the number of vortices. Second, when the coupling constant λ > 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.