The Gauss-Landau-Hall problem on Riemannian surfaces

In this paper we study the motion of a charged particle on a Riemannian surface under the influence of a positive magnetic field B. Using Moser's Twist Theorem and ideas from classical pertubation theory we find sufficient conditions to perpetually trap the motion of a particle with a sufficiently large charge in a neighborhood of a level set of the magnetic field. The conditions on the level set of the magnetic field that guarantee the trapping are local and hold near all no...

In this work, magnetic geodesics over the space of K\"ahler potentials are studied through a variational method for a generalized Landau-Hall functional. The magnetic geodesic equation is calculated in this setting and its relation to a perturbed complex Monge-Amp\`ere equation is given. Lastly, the magnetic geodesic equation is considered over the special case of toric K\"ahler potentials over toric K\"ahler manifolds.

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups.

In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is closed, on every energy level. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.

A magnetic field is defined by the property that its divergence is zero in a three dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to studies the magnetic flow asociated with the Killing magnetic field in a three dimensional oriented Riemann manifold, And then, we investigate the trajectories of the magnetic fields called as N-magn...

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $\Omega$ be a 2-form. We show that the magnetic flow of the pair $(g,\Omega)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $\Omega$ is a constant multiple of the area form of $g$ and the magnetic flow is a horocycle flow. This characterization of horocycle flows implies that if the magnetic flow of a pair $(g,\Omega)$ is $C^1$-conj...

We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solution...

In this paper we consider the gradient flow of the following Ginzburg-Landau type energy \[ F_\varepsilon(u) := \frac{1}{2}\int_{M}\vert D u\vert_g^2 +\frac{1}{2\varepsilon^2}\left(\vert u\vert_g^2-1\right)^2\mathrm{vol}_g. \] This energy is defined on tangent vector fields on a $2$-dimensional closed and oriented Riemannian manifold $M$ (here $D$ stands for the covariant derivative) and depends on a small parameter $\varepsilon>0$. If the energy satisfies proper bounds, wh...

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement th...

The quantum mechanics of a system of charged particles interacting with a magnetic field on Riemann surfaces is studied. We explicitly construct the wave functions of ground states in the case of a metric proportional to the Chern form of the $\theta$-bundle, and the wave functions of the Landau levels in the case of the the Poincar{\' e} metric. The degeneracy of the the Landau levels is obtained by using the Riemann-Roch theorem. Then we construct the Laughlin wave function...