Construction of many Hamiltonian stationary Lagrangian surfaces in Euclidean four-space

We make a large use of a Weierstrass representation formula to describe a variety of Hamiltonian stationary Lagrangian surfaces. Among the examples we give are the already known tori and cones, but also simply periodic cylinders, singularities of non-conical type and branch points of any order. Received: 11 November 2001 / Accepted: 23 January, 2002 / Published online: 5 September 2002     

The existence of Hamiltonian stationary Lagrangian tori in K?hler manifolds of any dimension

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They are natural generalizations of special Lagrangians or Lagrangian and minimal submanifolds. In this paper, we obtain a local condition that gives the existence of a smooth family of Hamiltonian stationary Lagrangian tori in K?hler manifolds. This criterion involves a weighted sum of holomorphic sectional curvatures. It can be considered as a complex analogue of the scalar curvature when the weighting are the same. The problem is also studied by Butscher and Corvino (Hamiltonian stationary tori in Kahler manifolds, 2008).     

Hamiltonian Stationary Tori in the Complex Projective Plane

Hamiltonian stationary Lagrangian surfaces are Lagrangian surfacesin a four-dimensional Kähler manifold which are criticalpoints of the area functional for Hamiltonian infinitesimaldeformations. In this paper we analyze these surfaces in thecomplex projective plane: in a previous work we showed thatthey correspond locally to solutions to an integrable system,formulated as a zero curvature on a (twisted) loop group. Herewe give an alternative formulation, using non-twisted loop groupsand, as an application, we show in detail why Hamiltonian stationaryLagrangian tori are finite type solutions, and eventually describethe simplest of them: the homogeneous ones. 2000 MathematicsSubject Classification 53C55 (primary), 53C42, 53C25, 58E12(secondary).     

Hamiltonian stationary tori in K?hler manifolds

A Hamiltonian stationary Lagrangian submanifold of a K?hler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a K?hler manifold of real dimension four to guarantee the existence of a family of small Hamiltonian stationary Lagrangian tori.     

Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions

We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus obtaining a first approach to a moduli space in terms of a simple algebraic-geometric problem on the plane. We also classify Hamiltonian stationary Klein bottles and show they self-intersect. Received: January 25, 2000.     

On a Lower Bound for the Energy Functional on a Family of Hamiltonian Minimal Lagrangian Tori in ℂ<Emphasis Type="Italic">P</Emphasis><Superscript>2</Superscript>

Under study is the energy functional on the set of Lagrangian tori in the complex projective plane. We prove that the value of the energy functional for a certain family of Hamiltonian minimal Lagrangian tori in the complex projective plane is strictly larger than for the Clifford torus.     

Hamiltonian Stationary Lagrangian Surfaces in <Emphasis Type="Italic">ℂP</Emphasis><Superscript>2</Superscript>

In this paper we use a new equivalent condition of Hamiltonian stationary Lagrangian surfaces in ℂP² to show that any Hamiltonian stationary Lagrangian torus in ℂP² can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspace of a certain loop Lie algebra, i.e., is of finite type. Mathematics Subject Classifications (2000): Primary 53C40; Secondary 53C42, 53D12     

Some families of special Lagrangian tori

 We give an explicit proof of the local version of Bryant's result [1], stating that any 3-dimensional real-analytic Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We then refine the theorem proving that a certain class of real-analytic one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. Two applications of these results show how the geometry of the moduli space of 3-dimesional special Lagrangian submanifolds differs considerably from the 2-dimensional one. First of all, applying Bryant's theorem and a construction due to Calabi we show that nearby elements of the local moduli space of a special Lagrangian 3-torus can intersect themselves. Secondly, we use our examples of one-parameter families to show that in dimension three (and higher) the moduli space of special Lagrangian tori is not, in general, special Lagrangian in the sense of Hitchin [13]. Received: 18 December 2001 / Revised version: 31 January 2002 / Published online: 16 October 2002 Mathematics Subject Classification (2000): 53-XX, 53C38     

Hamiltonian Stationary Lagrangian Surfaces in Euclidean Four-Space and Related Minimization Problems. Part II: Some Minimization Results

We study some minimization problems for Hamiltonian stationaryLagrangian surfaces in R⁴. We show that the flat Lagrangian torusS ¹ × S ¹ minimizes the Willmore functional among Hamiltonianstationary tori of its isotopy class, which gives a new proof of thefact that it is area minimizing in the same class. Considering theLagrangian flat cylinder as a torus in some quotient space R⁴/v Z, we show that it is also area minimizing in its isotopy class.     

An energy functional for Lagrangian tori in $$\mathbb {C}P^2$$

A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane \({\mathbb C}P^2\). Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional \(W^{-}\) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori.     

The classification of Hamiltonian stationary Lagrangian tori in {{\mathbb {CP}}^2} by their spectral data

It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in ${{\mathbb {CP}}^2}$ may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spectral data in a similar manner to the case of minimal Lagrangian tori in ${{\mathbb {CP}}^2}$ , but the details require a careful treatment of both the ??dressing construction?? and the spectral data to deal with a loop of flat connexions which is quadratic in the loop parameter.     

Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems

Chow et al. (J. Non. Sci. 12 (2002) 585) proved that the majority of the unperturbed tori on sub-manifolds will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems. The surviving tori might be elliptic, hyperbolic, or of mixed type.     

KAM theorems and open problems for infinite-dimensional Hamiltonian with short range

We introduce several KAM theorems for infinite-dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori.Especially,we introduce a KAM theorem by Yuan published in CMP(2002),which shows that there are rich KAM tori for a class of Hamiltonian with short range and with linearized operator of pure point spectra.We also present several open problems.     

Nontoric foliations by Lagrangian tori of toric Fano varieties

A construction of a foliation of a toric Fano variety by Lagrangian tori is presented; it is based on linear subsystems of divisor systems of various degrees invariant under the Hamiltonian action of distinguished function-symbols. It is shown that known examples of foliations (such as the Clifford foliation and D. Auroux’s example) are special cases of this construction. As an application, nontoric Lagrangian foliations by tori of two-dimensional quadrics and projective space are constructed.     

A compressible Hamiltonian electromagnetic gyrofluid model

A Lie-Poisson bracket is presented for a four-field gyrofluid model with magnetic field curvature and compressible ions, thereby showing the model to be Hamiltonian. The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants advected by distinct velocity fields. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.     

Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. We also consider the problem of existence of adiabatic invariants associated with a slowly perturbed Hamiltonian system. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1513–1528, November, 1999.     

Nonstandard Lagrangian tori and pseudotoric structures

We show that exotic Lagrangian tori constructed by Chekanov and Schlenk can be obtained for a large class of toric manifolds. For this, we translate their original construction into the language of pseudotoric structures. As an example, we construct exotic Lagrangian tori on a del Pezzo surface of degree six.     

Surgery on Lagrangian and Legendrian Singularities

Let be a smooth fiber bundle whose total space is a symplectic manifold and whose fibers are Lagrangian. Let L be an embedded Lagrangian submanifold of E. In the paper we address the following question: how can one simplify the singularities of the projection by a Hamiltonian isotopy of L inside E? We give an answer in the case when dim and both L and M are orientable. A weaker version of the result is proved in the higher-dimensional case. Similar results hold in the contact category.?As a corollary one gets an answer to one of the questions of V. Arnold about the four cusps on the caustic in the case of the Lagrangian collapse. As another corollary we disprove Y. Chekanov's conjecture about singularities of the Lagrangian projection of certain Lagrangian tori in . Submitted: January 1998, revised: January 1999.     

Hamiltonian reduced fluid model for plasmas with temperature and heat flux anisotropies

For an arbitrary number of species, we derive a Hamiltonian fluid model for strongly magnetized plasmas describing the evolution of the density, velocity, and electromagnetic fluctuations and also of the temperature and heat flux fluctuations associated with motions parallel and perpendicular to the direction of a background magnetic field. We derive the model as a reduction of the infinite hierarchy of equations obtained by taking moments of a Hamiltonian drift-kinetic system with respect to Hermite–Laguerre polynomials in velocity–magnetic-moment coordinates. We show that a closure relation directly coupling the heat flux fluctuations in the directions parallel and perpendicular to the background magnetic field provides a fluid reduction that preserves the Hamiltonian character of the parent drift-kinetic model. We find an alternative set of dynamical variables in terms of which the Poisson bracket of the fluid model takes a structure of a simple direct sum and permits an easy identification of the Casimir invariants. Such invariants in the limit of translational symmetry with respect to the direction of the background magnetic field turn out to be associated with Lagrangian invariants of the fluid model. We show that the coupling between the parallel and perpendicular heat flux evolutions introduced by the closure is necessary for ensuring the existence of a Hamiltonian structure with a Poisson bracket obtained as an extension of a Lie–Poisson bracket.     

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

Summary. Generalizing the degenerate KAM theorem under the Rüssmann nondegeneracy and the isoenergetic KAM theorem, we employ a quasilinear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth submanifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the submanifold, we shall show the following: (a) the majority of the unperturbed tori on the submanifold will persist; (b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; (c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a subisoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.