arXiv research

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits…

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…

All principal orbits of the standard Hamiltonian $T^n$-action on the complex projective space ${\mathbb C}P^n$ are Lagrangian tori.In this article, we prove that most of them are not volume minimizing under Hamiltonian isotopies of ${\mathbb C}P^n$ if the complex dimension $n$ is greater than two, although they are Ham…

This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.

Study of intersections in Hamiltonian orbits on cotangent bundles.

Abstract: Proves no non-trivial normal orbits for specific Hamiltonians.

In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ is H-stable when $n\leq 2$ and there exist infinitely many H-unst…

The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.

We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Ma…

In this paper we produce a lower bound for the number of periodic orbits of certain Hamiltonian vector fields near Bott-nondegenerate symplectic critical submanifolds. This result is then related to the problem of finding closed orbits of the motion of a charged low energy particle on a Riemannian manifold under the in…

Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M, and…

The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of coun…

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on …

The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided th…

Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation and so we induce on the orbits a simplectic form. By considering Hamiltonian systems on the orbits we study some features of them and finally find commuting functions under the corresponding Lie-Poisson brack…

Study of weighted nonlinear flags in symplectic geometry.

For a complex Lie group $G$ with a real form $G_0\subset G$, we prove that any Hamiltionian automorphism $φ$ of a coadjoint orbit $\mathcal O_0$ of $G_0$ whose connected components are simply connected, may be approximated by holomorphic $\mathcal O_0$-invariant symplectic automorphism of the corresponding coadjoint or…

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits …

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is $n$-dimensional and the symplectic manifold is $2n$-dimensional. In this case the $n$-dimen…

The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.

We prove that Hamiltonian characteristic classes defined as fibre integrals of powers of the coupling class are algebraically independent for generic coadjoint orbits.

The Hamiltonian flow of the standard metric Hamiltonian with respect to the twisted symplectic structure on the cotangent bundle describes the motion of a charged particle on the base. We prove that under certain natural hypotheses the number of periodic orbits on low energy levels for this flow is at least the sum of …

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the sub-Riemannian Heisenberg group. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. This system is known to admit closed orb…

We give an identification of the triple reduced product of three coadjoint orbits in SU(3) with a space of Hitchin pairs over a genus 0 curve with three punctures, where the residues of the Higgs field at the punctures are constrained to lie in fixed coadjoint orbits. Using spectral curves for the corresponding Hitchin…

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is giv…

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative perodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. In particular we prove that under appropriate hypothes…

Study a 10D symplectic manifold with 6 fixed points, linking to $G_2$ orbit.

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. More specifically we signific…

We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of prod…

Study of Hamiltonian flows on character varieties for self-intersecting curves.

We give a detailed construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.

We outline the construction of a proper C^2-smooth function on R^4 such that its Hamiltonian flow has no periodic orbits on at least one regular level set. This result can be viewed as a C^2-smooth counterexample to the Hamiltonian Seifert conjecture in dimension four.

Inspired by Katok's examples of Finsler metrics with a small number of closed geodesics, we present two results on Reeb flows with finitely many periodic orbits. The first result is concerned with a contact-geometric description of magnetic flows on the 2-sphere found recently by Benedetti. We give a simple interpretat…

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solu…

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is …

Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in six dimensions, and we use this correspondence to interpret symplectic fibrations between these orbits, and to analyse moment polytopes associated to the standard Hamiltonian torus action on the coadjoint orbits. The theory is then applied to d…

Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.

In this paper we compute the Reidemeister torsion of a isoenergetic surface for the integrable Hamiltonian system on the four-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds.

Study relates symplectic homology capacity to periodic orbits in Liouville domains.

For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^…

In this paper, we use Floer theory to study the Hofer length functional for paths of Hamiltonian diffeomorphisms which are sufficiently short. In particular, the length minimizing properties of a short Hamiltonian path are related to the properties and number of its periodic orbits.

The paper explores deformations of quasi-Hamiltonian spaces to Hamiltonian spaces.

In this paper we study a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a 3-manifold $M$ and prove the orbits of the constrained Hamiltonian dynamical system correspond to $G_2$-manifolds foliated by hypersurfaces diffeomorphic to $M\times \mathrm{SO}(3)$.

The paper calculates quantum cohomology for coadjoint orbits and Hamiltonian groups.

In 1993, Y.-G. Oh proposed a problem whether standard Lagrangian tori in C^n are volume minimizing under Hamiltonian isotopies of C^n. In this article, we prove that most of them do not have such property if the dimension n is greater than two. We also discuss the existence of Hamiltonian non-volume minimizing Lagrangi…

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Frechet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplecti…

We find coordinates, the metric tensor, the inverse metric tensor and the Laplace-Beltrami operator for the orbit space of Hamiltonian SU(2) gauge theory on a finite, rectangular lattice. This is done using a complete axial gauge fixing. The Gribov problem can be completely solved, with no remaining gauge ambiguities.