Periodic Magnetic Geodesics with Every Low Energy: Existence and Localization

Abstract

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.

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BibTeX RIS

BibTeX

@misc{assenza2026periodic,
  title = {{Periodic Magnetic Geodesics with Every Low Energy: Existence and Localization}},
  author = {Valerio Assenza and Gabriele Benedetti and Leonardo Macarini},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21931},
  eprint = {2609.21931},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21931},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Periodic Magnetic Geodesics with Every Low Energy: Existence and Localization
AU  - Valerio Assenza
AU  - Gabriele Benedetti
AU  - Leonardo Macarini
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21931
UR  - https://arxiv.org/abs/2609.21931
AB  - 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.
ER  -

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