arXiv subjects

Tom Stalljohann

Publications and source records attributed to Tom Stalljohann.

6 recordsLinked to original sources

arXiv2026

Periodic orbits for perturbations of Zoll contact forms

We consider the perturbation of a Zoll contact form on some prescribed domain of the underlying manifold by multiplying it with a positive function which is constant of value $1$ outside the domain. For every sufficiently $C^0$-small such perturbation we find a periodic Reeb orbit of the perturbed contact form intersecting the domain. As an application, starting from a Zoll Riemannian manifold, we demonstrate that for every exact magnetic field, with $C^0$-small magnetic potential vanishing outside some given domain, there exists a periodic magnetic geodesic intersecting this domain. The theorem is a rather direct consequence of a result which is of independent interest: For Hamiltonians sufficiently $C^0$-close to a defining Hamiltonian, we show existence of a gradient flow line of the corresponding Rabinowitz action functional with a constraint on the position of the cylinder component at $(0,0) \in \mathbb{R} \times \mathbb{S}^1$. This relies on a homotopy stretching argument for the Rabinowitz action functional, in the course of which we have to derive some delicate estimates to ensure compactness of the appearing moduli spaces.

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arXiv2026

An Abstract Perturbation Theorem for Compact Moduli Spaces

Given a compact zero set of a Fredholm section, our theorem guarantees the existence of a perturbed compact smooth manifold nearby, leaving the original zero set unaltered wherever transversality is already achieved. Such abstract perturbations allow for typical cobordism arguments. We illustrate this by re-proving a well-known theorem of Schwarz asserting the existence of critical points of the Hamiltonian action functional of different action values on symplectically aspherical manifolds.

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arXiv2026

Compactness of Moduli Spaces of Gradient Flow Lines in the Uniform Topology

We prove a compactness result for gradient flow lines in a general set-up which comprises both the situation of Morse gradient flow lines as well as Floer cylinders converging to a critical submanifold respectively. For the compactness result we have to impose two conditions. Both are readily verified in the Morse case but establishing the second condition in the Floer case poses a technical challenge and relies on an exponential decay estimate for Floer cylinders, with coefficient function continuously depending on the initial loop. This is a result of independent interest.

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arXiv2026

On the contact type conjecture for exact magnetic systems

In this article, we answer-for a class of magnetic systems-a question now known as the contact type conjecture, whose origin trace back to the 1998 work of Contreras, Iturriaga, Paternain, and Paternain. For a broad class of magnetic systems, we explicitly construct, on any closed manifold, an infinite-dimensional space of exact magnetic systems, which we refer to as magnetic systems of strong geodesic type. For each such system, there exists at least one null-homologous embedded periodic orbit on every energy level, with negative action for energies below the strict Mañé critical value. As a consequence, the corresponding energy surfaces are not of contact type below this threshold. Thus, for this class of systems, the contact type conjecture holds true. Moreover, for these systems, both the strict and the lowest Mañé critical values can be computed explicitly, and they coincide whenever the aforementioned periodic magnetic geodesic is contractible, without requiring any additional assumptions on the manifold. Several remarkable multiplicity results also hold, guaranteeing arbitrarily large numbers of embedded null-homologous periodic magnetic geodesics on every energy level. We illustrate the richness of this class through two types of examples. First, on any non-aspherical manifold, there exists a dense subset of the space of Riemannian metrics such that, for each such metric, one can construct an infinite-dimensional space of exact magnetic fields yielding magnetic systems of strong geodesic type. Second, on any closed contact manifold for which the strong Weinstein conjecture holds, one can construct an infinite-dimensional space of Riemannian metrics such that, for each such metric, the magnetic system induced by the fixed contact form is of strong geodesic type.

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arXiv2026

Topics in Magnetic Geometry: Interpolation, Intersections and Integrability

This paper develops new links between contact geometry, magnetic dynamics, and symmetry in exact magnetic systems. First, we establish an interpolation property for Killing magnetic systems on contact manifolds under an additional condition. Specifically, we show that the corresponding magnetic geodesic flow interpolates smoothly between the sub-Riemannian geodesic flow on the contact distribution and the flow of the vector field associated with a primitive of the magnetic field. Second, we show that Hamiltonian group actions associated with the magnetomorphism group produce Poisson-commuting integrals of motion for the magnetic flow. Finally, we obtain new structural results on totally magnetic submanifolds, showing that fixed-point sets of magnetomorphisms and intersections of totally magnetic submanifolds are again totally magnetic. The latter two results may be viewed as extensions of classical phenomena from Riemannian geometry to magnetic geometry.

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arXiv2025

The growth rate of closed prime magnetic geodesics on closed contact manifolds

In this paper, we prove that for any given closed contact manifold, there exists an infinite-dimensional space of Riemannian metrics which can be identified with the space of bundle metrics on the induced contact distribution. For each such metric, and for all energy levels, the number of embedded periodic orbits of the corresponding magnetic geodesic flow grows at least as fast as the number of geometrically distinct periodic Reeb orbits of period less than $t$. As a corollary, we deduce that for every closed 3-manifold which is not a graph manifold, there exists an open $C^1$-neighborhood of the set of nondegenerate contact forms such that for each contact form in this neighborhood, there exists an infinite-dimensional space of Riemannian metrics as above. For the corresponding magnetic systems, the number of prime closed magnetic geodesics grows at least exponentially on all energy levels. Consequently, the restriction of the magnetic geodesic flow to any energy surface has positive topological entropy.

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