arXiv research

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 clos…

The article finds non-trivial Zoll magnetic systems for surfaces of any genus.

Normal forms prove dynamical results for magnetic fields on surfaces.

Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.

Constructs smooth integrable magnetic systems on a two-torus.

We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close t…

We prove that for Matveev and Shevchishin superintegrable system, with a linear and a cubic integral, the metrics defined on S^2 and on Tannery's orbifold T^2 are either Zoll or Tannery metrics.

The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $Ω$ be an odd-symplectic form on an oriented closed manifold $Σ$ of odd dimension. We say that $Ω$ is Zoll if the trajectories of the flow given …

Let $Σ$ be a connected closed three-manifold, and let $t_Σ$ be the order of the torsion subgroup of $H_1(Σ;\mathbb Z)$. For a contact form $α$ on $Σ$, we denote by $\mathrm{Volume}(α)$ the contact volume of $α$, and by $T_{\min}(α)$ and $T_{\max}(α)$ the minimal period and the maximal period of prime periodic orbits of…

Almost Zoll affine surface found on cylinder.

Study proves certain Zoll manifolds with entire Grauert tubes are isometric.

New index characterizes non-smooth Zoll convex bodies.

Zoll manifolds with entire Grauert tubes are proven to be standard complex projective spaces.

Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian Möbius strips of non constant curvature all of whose spacelike geodesics are cl…

Optimal Strichartz estimates for Schrödinger on Zoll manifolds.

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

Extends E. Hopf's theorem to magnetic systems without conjugate points.

This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.

We strengthen our previous results regarding the moduli spaces of Zoll metrics and Zoll projective structures on S^2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding of RP^2 in CP_2 arises from a unique Zoll projective structure on the 2-sphere. Our method…

The isospectral problem for p-widths is solved using Zoll metrics on S^2.

New Zoll families of minimal spheres found in spheres and projective spaces.

Study shows finiteness of magnetic hypersurfaces on closed manifolds.

The paper explores how magnetic systems' spectra can identify metrics and 1-forms.

Study of Lévy flights on Zoll surfaces, revealing geometric information.

Researchers prove integrability of magnetic systems on spheres up to dimension 6.

New findings on magnetic geodesic flows and periodic motions.

Metrics are isometric for certain Anosov magnetic systems.

Study magnetic curvature on Lie groups, extending Milnor's work.

The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.

Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.

Study reveals new geometric structures for magnetic field Hamiltonian systems.

Researchers solve boundary and scattering rigidity problems for magnetic systems.

Study Zoll manifolds with boundary, showing unique geodesic properties.

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

Study magnetic geodesics on odd spheres, computing critical energy values.

In this paper, we study the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.

Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.

The question of whether or not the set of Zoll metrics on the 2-sphere is connected is still open. Here we show that a naive application of the Ricci flow is not sufficient to answer this problem.

Let $α$ be a contact form on a connected closed three-manifold $Σ$. The systolic ratio of $α$ is defined as $ρ_{\mathrm{sys}}(α):=\tfrac{1}{\mathrm{Vol}(α)}T_{\min}(α)^2$, where $T_{\min}(α)$ and $\mathrm{Vol}(α)$ denote the minimal period of periodic Reeb orbits and the contact volume. The form $α$ is said to be Zoll …

Study of gyroscopic Chaplygin systems and magnetic flows on spheres.

We consider non-self-adjoint Schrödinger operators $Δ+V$ where $Δ$ is the Laplace-Beltrami operator on a Zoll manifold $X$ and $V\in C^\infty(X,\mathbb C)$. We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.

Challenge to separate Earth's magnetic field from vehicle's magnetic field for accurate navigation.

A controlled magnetic Hamiltonian (CMH) system is a regular controlled Hamiltonian (RCH) system with magnetic symplectic form, it is an important special case of RCH system. Note that there is a magnetic term on the cotangent bundle of the Heisenberg group, such that we can define a CMH system with symmetry of the Heis…

The systolic ratio of a contact form $α$ on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where $T_{\min}(α)$ is the minimal period of closed Reeb orbits on $(S^3,α)$. A Zoll contact form is a contact form such that all the orbits of the corresponding R…

The paper constructs metrics on spheres with families of minimal hypersurfaces.

Examples are presented of how the geometric notion of the mean curvature is used for general magnetic field configurations and magnetic surfaces. It is shown that the mean magnetic curvature is related to the variation of the absolute value of the magnetic field along its lines. Magnetic surfaces of constant mean curva…

MagNet uses neural networks to predict multi-agent dynamics from observations.

We find necessary and sufficient conditions for a local geodesic flow of an affine connection on a surface to admit a linear first integral. The conditions are expressed in terms of two scalar invariants of differential orders 3 and 4 in the connection. We use this result to find explicit obstructions to the existence …