Exotic smooth structures and symplectic forms on closed manifolds

In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.     

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.     

An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in {\mathbb C}^2 related to Oh's conjecture

We compute loops integrals on Hamiltonian stationary Lagrangian tori in which are symplectic invariants, then we show an isoperimetric inequality involving these invariants and the area. Finally, we show that the flat torus has least area among Hamiltonian stationary Lagrangian tori of its isotopy class. Received: 4 December 2000; in final form: 18 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).     

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.     

Examples of unstable Hamiltonian-minimal Lagrangian tori in C2

A new family of Hamiltonian-minimal Lagrangian tori in the complex Euclidean plane is constructed. They are the first known unstable ones and are characterized in terms of being the only Hamiltonian-minimal Lagrangian tori (with non-parallel mean curvature vector) in C² admitting a one-parameter group of isometries.     

Pseudotoric structures on toric symplectic manifolds

We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds.     

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.     

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.     

Minimal Lagrangian 2-Tori in CP2 Come in Real Families of Every Dimension

It is shown that for every non-negative integer n, there isa real n-dimensional family of minimal Lagrangian tori in CP²,and hence of special Lagrangian cones in C³ whose link is atorus. The proof utilises the fact that such tori arise fromintegrable systems, and can be described using algebro-geometric(spectral curve) data.     

Diffeomorphisms and Almost Complex Structures on Tori

We prove that there exist diffeomorphisms of tori, supported in a disc, which are not isotopic to symplectomorphisms with respect to the standard symplectic structure. This yields a partial negative answer to a question of Benson and Gordon about the existence of symplectic structures on tori with exotic differential structure. Mathematics Subject Classifications (2000): 53C15, 53D35.     

Semiclassical second microlocal propagation of regularity and integrable systems

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold X with respect to a Lagrangian submanifold of T ^* X. The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of T ^*? ⁿ , and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to Hörmander’s theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g., eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.     

KAM theory in configuration space

A new approach to the Kolmogorov-Arnold-Moser theory concerning the existence of invariant tori having prescribed frequencies is presented. It is based on the Lagrangian formalism in configuration space instead of the Hamiltonian formalism in phase space used in earlier approaches. In particular, the construction of the invariant tori avoids the composition of infinitely many coordinate transformations. The regularity results obtained are applied to invariant curves of monotone twist maps. The Lagrangian approach has been prompted by a recent study of minimal foliations for variational problems on a torus by J. Moser. This research has been supported by the Nuffields Foundation under grant SCI/180/173/G and by the Stiftung Volkswagenwerk.     

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     

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.     

Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: Given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities, we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer-assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to \(\varepsilon =0.9716\)), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.     

Semiclassical maslov asymptotics with complex phases. I. General approach

A method of constructing semiclassical asymptotics with complex phases is presented for multidimensional spectral problems (scalar, vector, and with operator-valued symbol) corresponding to both classically integrable and classically nonintegrable Hamiltonian systems. In the first case, the systems admit families of invariant Lagrangian tori (of complete dimension equal to the dimensionn of the configuration space) whose quantization in accordance with the Bohr—Sommerfeld rule with allowance for the Maslov index gives the semiclassical series in the region of large quantum numbers. In the nonintegrable case, families of Lagrangian tori with complete dimension do not exist. However, in the region of regular (nonchaotic) motion, such systems do have invariant Lagrangian tori of dimensionk (incomplete dimension). The construction method associates the families of such tori with spectral series covering the region of intermediate quantum numbers. The construction includes, in particular, new quantization conditions of Bohr—Sommerfeld type in which other characteristics of the tori appear instead of the Maslov index. Applications and also generalizations of the theory to Lie groups will be presented in subsequent publications of the series.Moscow Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 2, pp. 215–254, August, 1992.     

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 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).