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arXiv 2502.01151math-phmath.MP
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弯曲空间中磁性 Ginzburg-Landau 理论的涡旋

Vortices for the magnetic Ginzburg-Landau theory in curved space

Lei Cao, Yilu Xu, Shouxin Chen

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AI总结：

本文研究了弯曲空间中与 Einstein 方程耦合的磁性 Ginzburg-Landau 理论的涡旋解，分别在自对偶（λ=1）和径向对称（λ>0）情形下，利用约束极小化、单调迭代及打靶-不动点方法证明了解的存在性，并阐明了涡旋数量对量子化磁通量与总曲率的决定作用。

摘要：

由于 Ginzburg-Landau 理论关注宏观现象，且引力在宏观层面影响物体的相互作用方式，因此在弯曲空间（即存在引力的情况下）中研究 Ginzburg-Landau 理论变得具有重要意义。本文针对与 Einstein 方程耦合的磁性 Ginzburg-Landau 理论的涡旋解建立了一些存在性定理。首先，当耦合常数 λ=1 时，我们从 Ginzburg-Landau 理论中得到一种自对偶结构，随后从耦合系统中推导出一个带有幂型奇点引力项的偏微分方程。为了克服涡旋处奇点阶数带来的困难，采用了约束极小化方法和单调迭代方法。我们还证明了量子化磁通量和总曲率由涡旋的数量决定。其次，当耦合常数 λ>0 时，我们使用一个合适的拟设（ansatz）来得到弯曲空间中磁性 Ginzburg-Landau 理论的径向对称情形。通过结合两步迭代打靶论证和不动点定理方法，获得了对称涡旋解的存在性。通过应用一系列分析技术，确立了解的一些基本性质。

英文摘要：

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.

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