The Gauss-Landau-Hall problem on Riemannian surfaces

The aim of this work is the study of magnetic trajectories on nilmanifolds. The magnetic equation is written and the corresponding solutions are found for a family of invariant Lorentz forces on a 2-step nilpotent Lie group equipped with a left-invariant metric. Some examples are computed in the Heisenberg Lie groups $H_n$ for $n=3,5$, showing differences with the case of exact forms. Interesting magnetic trajectories related to elliptic integrals appear in $H_3$. The questio...

We consider billiard ball motion in a convex domain of a constant curvature surface influenced by the constant magnetic field. We prove that if the billiard map is totally integrable then the boundary curve is necessarily a circle. This result is a manifestation of the so-called Hopf rigidity phenomenon which was recently obtained for classical billiards on constant curvature surfaces.

Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields $X$ on arbitrary Riemann surfaces $M$. By vector field singularities we understand zeros, poles, isolated essential singularities and accumulation points of the above kind. In this framework, a singular analytic vector field $X$ has canonically associated; a 1-form, a quadratic differential, a flat metric (with a geodesic foliation), a glo...

The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, for parametrized curves in Lagrange Grassmannians satisfying very general assumptions. The stru...

The main result presented here is that the flow associated with a riemannian metric and a non zero magnetic field on a compact oriented surface without boundary, under assumptions of hyperbolic type, cannot have the same length spectrum of topologically corresponding periodic orbits as the geodesic flow associated with another riemannian metric having a negative curvature and the same total volume. The main tool is a regularization inspired by U. Hamenst\"adt's methods.

Let $M$ be a closed surface and let $\{g_s \ | \ s \in (-\epsilon, \epsilon)\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{\kappa_s : M \rightarrow \mathbb{R} \ | \ s \in (-\epsilon, \epsilon)\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s, \kappa_s) \ | \ s \in (-\epsilon, \epsilon)\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s \in (-\epsilon, ...

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ${\mathbb{R}}^{2}$ in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding...

This is an introductory chapter in a series in which we take a systematic study of the Yang-Mills equations on curved space-times. In this first, we provide standard material that consists in writing the proof of the global existence of Yang-Mills fields on arbitrary curved space-times using the Klainerman-Rodnianski parametrix combined with suitable Gr\"onwall type inequalities. While the Chru\'sciel-Shatah argument requires a simultaneous control of the $L^{\infty}_{loc}$ a...

To a Riemannian manifold ${(M, g)}$ endowed with a magnetic form ${\sigma}$ and its Lorentz operator ${\Omega}$ we associate an operator ${M^{\Omega}}$, called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric ${g}$ together with terms of perturbation due to the magnetic interaction of ${\sigma}$. From ${M^{\Omega}}$ we derive the magnetic sectional curvature ${\mathrm{Sec}^{\Omega}}$ and the magnetic Ricci curvature $...

The quantum dynamics of a two-dimensional charged spin $1/2$ particle is studied for general, symmetry--free curved surfaces and general, nonuniform magnetic fields that are, when different from zero, orthogonal to the defining two surface. Although higher Landau levels generally lose their degeneracy under such general conditions, the lowest Landau level, the ground state, remains degenerate. Previous discussions of this problem have had less generality and/or used supersymm...