Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.06951

On the Rigidity of Hamiltonians which are Zoll Near a Minimum, with an Application to Magnetic Systems and Almost-K\"ahler Manifolds

Gabriele Benedetti·Johanna Bimmermann·Samanyu Sanjay
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study Hamiltonian systems near a compact symplectic Morse-Bott minimum. Our first result shows that if the flow is Zoll (that is, it induces a free circle action) along a sequence of energy levels converging to the minimum, then the Hessian of the Hamiltonian in the symplectic normal directions must be compatible with the restriction of the symplectic structure to the normal bundle (that is, its representing endomorphism is a complex structure of the symplectic normal bundle). For our second result, we specialize to magnetic systems on closed manifolds with symplectic magnetic form. In this setting, if the system is Zoll along a sequence of energy levels converging to the minimum, then the metric is compatible with the magnetic form and therefore defines an almost K\"ahler structure. We show that a natural curvature quantity, consisting of the holomorphic sectional curvature corrected by a term measuring the non-integrability of the almost complex structure, must be constant. In particular, we obtain a dynamical characterization of complex space forms among K\"ahler manifolds. Together, these results establish strong rigidity of systems which are Zoll at energies close to a Morse-Bott minimum, in the symplectic and in the magnetic settings.

Explore related subjects

math.SGmath.DGmath.DS

Keep this discovery

BibTeXRIS

Gabriele Benedetti, Johanna Bimmermann, Samanyu Sanjay. 2026-04-08. On the Rigidity of Hamiltonians which are Zoll Near a Minimum, with an Application to Magnetic Systems and Almost-K\"ahler Manifolds. https://arxiv.org/abs/2604.06951

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extended Future Tube Conjecture for Unipotent Subgroups

Let $\Omega$ be the Lorentz future cone in $\mathbb{R}^{d+1}$ with respect to the Lorentz product and let $T^M$ be the $M$-fold product of the future tube $T=\mathbb{R}^{d+1}+i\Omega$. The Lorentz group $\mathrm{SO}_0(1,d)$ acts diagonally on $T^M$, and its complexification $\mathrm{SO}(1,d)^\mathbb{C}$ acts on $\mathbb{C}^{(d+1)\times M}$. We prove that the domain $G^\mathbb{C}\cdot T^M$ is a Stein manifold for any connected unipotent subgroup $G$ of $\mathrm{SO}_0(1,d)$.

math.SG↗
arXiv2026

The First Correction Term in the Asymptotic Expansion of Bohr--Sommerfeld Lagrangian States

Let $\Lambda$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact K\"ahler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient K\"ahler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

math.SG↗
arXiv2026

Classification of Legendrian doubles and suspensions

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

math.SG↗