arXiv subjects

Valerio Assenza

Publications and source records attributed to Valerio Assenza.

5 recordsLinked to original sources

arXiv2026

Periodic Magnetic Geodesics with Every Low Energy: Existence and Localization

Let $(Q,g)$ be a Riemannian manifold equipped with a non-identically zero magnetic field represented by a closed $2$-form $β$. We allow $Q$ to be non-compact and do not assume that the metric $g$ is complete. We prove that if the magnetic strength, defined as the pointwise norm of $β$, attains a strict local maximum on a non-empty compact set $K$, then every sufficiently low energy level carries a contractible periodic magnetic geodesic of the pair $(g,β)$ localized near $K$. More precisely, such orbits exist in every neighborhood of $K$, and their lengths converge to zero with the energy. In particular, if $Q$ is compact, then every sufficiently small energy level carries a contractible periodic magnetic geodesic. We also show that, in general, neither the non-emptiness nor the compactness of $K$ can be omitted. Our proof relies on the calculus of variations of the Lagrangian action functional, and uses several new ideas. More precisely, we overcome: (i) the non-exactness of $β$ by restricting the minimax to the set of short loops; (ii) the non-completeness of $g$ by combining a compactification of $Q$ with Thom's Jet Transversality and a blow-up for sequences of magnetic geodesics with energy tending to zero; (iii) the possible non-compactness of Palais--Smale sequences by the positivity of the Ricci magnetic curvature for low energy established by the first-named author. Unlike previous work, Struwe's monotonicity argument cannot be used for our purposes, and we rely on a two-Lyapunov-function argument due to Abbondandolo and Majer.

math.SG↗

arXiv2026

Electromagnetic curvature via Jacobi-Maupertuis and beyond

In the setting of electromagnetic systems, we propose a new definition of electromagnetic Ricci curvature, naturally derived via the classical Jacobi-Maupertuis reparametrization from the recent works of Assenza [IMRN, 2024] and Assenza, Marshall Reber, Terek [Communications in Mathematical Physics, 2025]. On closed manifolds, we show that if the magnetic force is nowhere vanishing and the potential is sufficiently small in the $C^2$ norm, then this Ricci curvature is positive for energies close to the maximum value of the potential $e_0$. As a main application, under these assumptions, we extend the existence of contractible closed orbits at energy levels near $e_0$ from almost every to everywhere.

math.DG↗

arXiv2024

Magnetic flatness and E. Hopf's theorem for magnetic systems

Using the notion of magnetic curvature recently introduced by the first author, we extend E. Hopf's theorem to the setting of magnetic systems. Namely, we prove that if the magnetic flow on the s-sphere bundle is without conjugate points, then the total magnetic curvature is non-positive, and vanishes if and only if the magnetic system is magnetically flat. We then prove that magnetic flatness is a rigid condition, in the sense that it only occurs when either the magnetic form is trivial and the metric is flat, or when the magnetic system is Kähler, the metric has constant negative sectional holomorphic curvature, and s equals the Mañé critical value.

math.DG↗

arXiv2024

Marked length spectrum rigidity for Anosov magnetic surfaces

We show that if $M$ is a closed, connected, oriented surface, and two Anosov magnetic systems on $M$ are conjugate by a volume-preserving conjugacy isotopic to the identity, with their magnetic forms in the same cohomology class, then the metrics are isometric. This extends the recent result by Guillarmou, Lefeuvre, and Paternain to the magnetic setting.

math.DG↗

arXiv2024

Magnetic curvature and existence of a closed magnetic geodesic on low energy levels

To a Riemannian manifold $(M, g)$ endowed with a magnetic form $σ$ and its Lorentz operator $Ω$ we associate an operator $M^Ω$, 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 $σ$. From $M^Ω$ we derive the magnetic sectional curvature $Sec^Ω$ and the magnetic Ricci curvature $Ric^Ω$ which generalize in arbitrary dimension the already known notion of magnetic curvature previously considered by several authors on surfaces. On closed manifolds, under the assumption of $Ric^Ω$ being positive on an energy level below the Mañé critical value, with a Bonnet-Myers argument, we establish the existence of a contractible periodic orbit. In particular, when $σ$ is nowhere vanishing, this implies the existence of a contractible periodic orbit on every energy level close to zero. Finally, on closed oriented even dimensional manifolds, we discuss about the topological restrictions which appear when one requires $Sec^Ω$ to be positive.

math.SG↗