Vortices for the magnetic Ginzburg-Landau theory in curved space
Lei Cao, Yilu Xu, Shouxin Chen
TL;DR
This work analyzes the magnetic Ginzburg–Landau theory coupled to Einstein gravity in curved space, deriving a self‑dual reduction at λ = 1 and reducing the problem to a nonlinear elliptic PDE for u = abla ext{ln}|\phi|^2 with an explicit gravitational factor e^{ exteta}. It proves existence of multi‑vortex solutions for finite total vortex number under the deficit constraint 4π G N ≤ 1, using constrained variational methods and monotone iterations, and characterizes the quantized flux, energy, and Gauss curvature contributions tied to N. For general λ > 0, it establishes the existence of radially symmetric N‑vortex solutions via a two‑step shooting method combined with a Schauder fixed‑point argument, and provides precise asymptotic and monotonicity properties. Together, these results reveal how gravity and curvature shape vortex structures, flux quantization, and geometric deficits in the Ginzburg–Landau framework. The findings have implications for macroscopic superconducting phenomena in gravitational backgrounds and broaden the mathematical understanding of vortex solutions in curved spaces.
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 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 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.