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Electromagnetic curvature via Jacobi-Maupertuis and beyond

Valerio Assenza, Giorgia Testolina

TL;DR

The paper develops a perturbative framework for electromagnetic dynamics on closed manifolds by introducing an electromagnetic curvature via the Jacobi–Maupertuis reparametrization. It provides an explicit formula for the extended curvature $\mathrm{Ric}^{g,\sigma,U}_k$ and proves that, when the magnetic form is nowhere vanishing and the potential $U$ is small in $C^2$, this curvature is positive for energies near $e_0$, yielding a perturbative existence result for contractible closed $(g,\sigma,U)$-geodesics for energies $(e_0,\nu_0]$. The analysis combines a detailed conformal metric change $g_k=2(k-U)g$, a Struwe-type minimax variational scheme with a Bonnet–Myers type bound to control periods, and a careful surface case that highlights the necessity of $C^2$-smallness. Collectively, these results extend known magnetic-case results to a broad electromagnetic setting and establish a principled, curvature-driven mechanism for the appearance of closed orbits near the top of the potential energy landscape.

Abstract

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.

Electromagnetic curvature via Jacobi-Maupertuis and beyond

TL;DR

The paper develops a perturbative framework for electromagnetic dynamics on closed manifolds by introducing an electromagnetic curvature via the Jacobi–Maupertuis reparametrization. It provides an explicit formula for the extended curvature and proves that, when the magnetic form is nowhere vanishing and the potential is small in , this curvature is positive for energies near , yielding a perturbative existence result for contractible closed -geodesics for energies . The analysis combines a detailed conformal metric change , a Struwe-type minimax variational scheme with a Bonnet–Myers type bound to control periods, and a careful surface case that highlights the necessity of -smallness. Collectively, these results extend known magnetic-case results to a broad electromagnetic setting and establish a principled, curvature-driven mechanism for the appearance of closed orbits near the top of the potential energy landscape.

Abstract

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 norm, then this Ricci curvature is positive for energies close to the maximum value of the potential . As a main application, under these assumptions, we extend the existence of contractible closed orbits at energy levels near from almost every to everywhere.

Paper Structure

This paper contains 12 sections, 11 theorems, 113 equations.

Table of Contents

  1. Introduction and results
  2. Electromagnetic dynamics
  3. Electromagnetic curvature
  4. Explicit formula for $\mathrm{Ric}^{g,\sigma,U}$ and proof of Theorem \ref{['thm_positivity']}
  5. Electromagnetic curvature on surfaces
  6. Existence of a closed $(g,\sigma,U)$-geodesic: proof of Theorem \ref{['thm_existence']} (and Theorem \ref{['TheroemA1']})
  7. Variational Setting
  8. A Bonnet-Myers type argument (see assenza)
  9. Proof of Theorem \ref{['thm_existence']} (and Theorem \ref{['TheroemA1']})
  10. Struwe monotonicity argument
  11. Weakly exact case
  12. Non weakly exact case

Key Result

Theorem 1.1

Let $\sigma$ be a nowhere vanishing magnetic form on a closed manifold $M$. Then, for every metric $g$, we can find a $C^2$-neighborhood $\mathcal{V}$ of the identically zero function such that, if $U \in \mathcal{V}$, there exists $\nu_0 \in (e_0, c]$ with the property that, for every $k \in (e_0,

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm_positivity']}
  • Example 2.3
  • Lemma 3.1
  • ...and 12 more