# On the bifurcation theory of the Ginzburg-Landau equations

@inproceedings{Nagy2022OnTB, title={On the bifurcation theory of the Ginzburg-Landau equations}, author={'Akos Nagy and Gonccalo Oliveira}, year={2022} }

A BSTRACT . We construct nonminimal and irreducible solutions to the Ginzburg–Landau equations on closed manifolds of arbitrary dimension with trivial ﬁrst real cohomology. Our method uses bifurcation theory where the “bifurcation points” are characterized by the eigenvalues of a Laplace-type operator. To our knowledge these are the ﬁrst such examples on nontrivial line bundles.

## References

SHOWING 1-10 OF 14 REFERENCES

### Invariant connections and vortices

- Mathematics
- 1993

We study the vortex equations on a line bundle over a compact Kähler manifold. These are a generalization of the classical vortex equations over ℝ2. We first prove an invariant version of the theorem…

### Vortices in holomorphic line bundles over closed Kähler manifolds

- Mathematics
- 1990

We apply a modified Yang-Mills-Higgs functional to unitary bundles over closed Kähler manifolds and study the equations which govern the global minima. The solutions represent vortices in holomorphic…

### Convergence of the self-dual $U(1)$-Yang-Mills-Higgs energies to the $(n-2)$-area functional

- Mathematics
- 2021

Given a hermitian line bundle L → M on a closed Riemannian manifold (M, g), the self-dual Yang–Mills–Higgs energies are a natural family of functionals

### Bifurcation Theory of Functional Differential Equations

- Mathematics
- 2013

Introduction to Dynamic Bifurcation Theory.- Introduction to Functional Differential Equations.-Center Manifold Reduction.- Normal form theory.- Lyapunov-Schmidt Reduction.- Degree theory.-…

### Irreducible Ginzburg–Landau Fields in Dimension 2

- Mathematics
- 2016

Ginzburg–Landau fields are the solutions of the Ginzburg–Landau equations which depend on two positive parameters, $$\alpha $$α and $$\beta $$β. We give conditions on $$\alpha $$α and $$\beta $$β for…

### Uhlenbeck Compactness

- Mathematics

Preface The origin of this book lies at the beginning of my graduate studies, when I just could not understand Uhlenbeck compactness, let alone see whether it would also hold for my cases – on…

### Minimal submanifolds from the abelian Higgs model

- MathematicsInventiones mathematicae
- 2020

Given a Hermitian line bundle L→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}…

### Stable Solutions to the Abelian Yang–Mills–Higgs Equations on $$S^2$$ and $$T^2$$

- MathematicsThe Journal of Geometric Analysis
- 2020

We show under natural assumptions that stable solutions to the abelian Yang--Mills--Higgs equations on Hermitian line bundles over the round $2$-sphere actually satisfy the vortex equations, which…