Title: Vortices for the magnetic Ginzburg-Landau theory in curved space Show Chinese title

Title: 曲线空间中的磁Ginzburg-Landau理论的涡旋

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. Show Chinese abstract

Abstract: 由于Ginzburg-Landau理论涉及宏观现象，而引力影响物体在宏观层面的相互作用。 因此，研究在弯曲空间中的Ginzburg-Landau理论（即存在引力的情况下）变得相关。 本文建立了与爱因斯坦方程耦合的磁Ginzburg-Landau理论的涡旋解的一些存在定理。 首先，当耦合常数\lambda=1, we 获得来自Ginzburg-Landau理论的自对偶结构时，从耦合系统中推导出一个带有引力项且具有幂次型奇点的偏微分方程。 为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。 我们还证明了量化通量和总曲率由涡旋的数量决定。 其次，当耦合常数\lambda >0 时，我们使用适当的假设得到弯曲空间中磁Ginzburg-Landau理论的径向对称情况。 通过对两个步骤的迭代射击论证和不动点定理方法相结合，获得了对称涡旋解的存在性。 通过应用一系列分析技术，建立了这些解的一些基本性质。