Existence d'orbites périodiques complètement elliptiques des Hamiltoniens convexes présentant certaines symétries

Using certain quadratic forms associated to symplectic endomorphisms which we compare with the Clarke-Ekeland dual action functional, we prove: THEOREM. — Let H be a C²-Hamiltonian defined on R²ⁿ, strictly convex, proper and invariant under a certain symplectic rational positive and non-degenerate rotation (this is defined in the introduction); then, every hypersurface of H contains a completely elliptic periodic orbit. This generalizes the result of G. Dell'Antonio, B. D'Onofrio and I. Ekeland contained in [1].     

Un principe variationnel pour les variétés symplectiques

Given a symplectic manifold (M, ω) and a function H : M → R, we construct an action functional A on paths in M with values in a torus T^k. We then show that a path is a solution of Hamilton's equations if and only if it is a critical point for A.     

Symplectic embeddings of ellipsoids

We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr _n ² ≤2r ₁ ² the symplectic ellipsoidE(r ₁,…,r _n)with radiir ₁≤…≤r _ndoes not symplectically embed into a ball of radius strictly smaller thanr _n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.     

Symplectically hyperbolic manifolds

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are:

• If a symplectic form represents a bounded cohomology class then it is hyperbolic.

• The symplectic hyperbolicity is equivalent to a certain isoperimetric inequality.

• The fundamental group of symplectically hyperbolic manifold is non-amenable.

We also construct hyperbolic symplectic forms on certain bundles and Lefschetz fibrations, discuss the dependence of the symplectic hyperbolicity on the fundamental group and discuss some properties of the group of symplectic diffeomorphisms of a symplectically hyperbolic manifold.

Densest Geodesic Ball Packings to <Emphasis Type="Bold">S</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Bold">R</Emphasis> space groups generated by screw motions

In this paper we study the locally optimal geodesic ball packings with equal balls to the S ² × R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive geodesic ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S ² × R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnár has shown in [9] that the Thurston geometries have an unified interpretation in the real projective 3-sphere \({\mathcal{PS}^3}\). In our work we shall use this projective model of S ² × R geometry.     

Spherical Lagrangians via ball packings and symplectic cutting

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, $S^{2}$ or $\mathbb{RP }^{2}$ , in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction, this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.     

Groupes d'automorphismes et plongements symplectiques de boules dans les variétés rationnellesAutomorphism groups and embeddings of symplectic balls into rational manifolds

We study the space Pl(c,λ) of symplectic embeddings of the closed ball $\text{B}{}^4\text{(c)?}\text{R}{}^4$ of capacity c in (S²×S²,(1+λ)ω_st⊕ω_st). When λ=0, we show that this space behaves like the space of ordinary differential embeddings and hence that its homotopy type does not depend on c. When λ>0, we prove that the restriction Pl(c′,λ)→Pl(c,λ) is no longer a homotopy equivalence when c and c′ lie on different sides of the value λ. To cite this article: F. Lalonde, M. Pinsonnault, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 931–934.     

Toric symplectic ball packing

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion.In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory.Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.     

Lagrangian barriers and symplectic embeddings

We prove that every symplectic Kähler manifold (M;W) (M;\Omega) with integral [W] [\Omega] decomposes into a disjoint union (M,W) = (E,w₀) \coprod D (M,\Omega) = (E,\omega_0) \coprod \Delta , where (E,w₀) (E,\omega_0) is a disc bundle endowed with a standard symplectic form w₀ \omega_0 and D \Delta is an isotropic CW-complex. We perform explicit computations of this decomposition on several examples.¶As an application we establish the following symplectic intersection phenomenon: There exist symplectically irremovable intersections between contractible domains and Lagrangian submanifolds. For example, we prove that every symplectic embedding j:B²ⁿ(l) ? \Bbb CPⁿ \varphi:B^{2n}(\lambda) \to {\Bbb C}P^n of a ball of radius l² 3 1/2 \lambda^2 \ge 1/2 must intersect the standard Lagrangian real projective space \Bbb RPⁿ ì \Bbb CPⁿ {\Bbb R}P^n \subset {\Bbb C}P^n .     

An integral estimate and the equivalent norms on F(p,q, s,k) spaces in the unit ball

In this article, the authors give a typical integral's bidirectional estimates for all cases. At the same time, several equivalent characterizations on the F(p, q, s, k) space in the unit ball of Cⁿ are given.     

Geometric Anosov flows of dimension 5

We show that for a smooth Anosov flow on a closed five dimensional manifold, if it has C^∞ Anosov splitting and preserves a C^∞ pseudo-Riemannian metric, then up to a special time change and finite covers, it is C^∞ flow equivalent either to the suspension of a symplectic hyperbolic automorphism of $\text{T}{}^4$, or to the geodesic flow on a three dimensional hyperbolic manifold. To cite this article: Y. Fang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A new linear quotient of C 4 admitting a symplectic resolution

We show that the quotient C ⁴/G admits a symplectic resolution for ${G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}$ . Here Q ₈ is the quaternionic group of order eight and D ₈ is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements ?Id of each. It is equipped with the tensor product representation ${{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}$ . This group is also naturally a subgroup of the wreath product group ${Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}$ . We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C ⁴/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.     

Essential unitarization of symplectics and applications to field quantization

Symplectic operators satisfying generic and group-invariant (spectral) positivity conditions are studied; the theory developed is applied and illustrated to determine the unique invariant frequency decomposition (equivalently, linear quantization with invariant vacuum state) of the Klein-Gordon equation in non-static spacetimes. Let (H, Ω) be any linear topological symplectic space such that there exists a real-linear and topological isomorphism of H with some complex Hilbert space carrying Ω into the imaginary part of the scalar product. Then any bounded invertible symplectic S ∈ Sp(H) (resp. bounded infinitesimally symplectic A ∈ sp(H)) which satisfies $\text{Ω(Sv, v) > 0}$ (resp. $\text{Ω(Av, v) > 0}$) for all nonzero v ω H, where S + I is invertible, is realized uniquely and constructively as a unitary (resp. skewadjoint) operator in a complex Hilbert space which depends in general on the operator and typically only densely intersects H. The essentially unique weakly and uniformly closed invariant convex cones in sp(H) are determined, extending previously known results in the finite-dimensional case. A notion of “skew-adjoint extension” of a closed semi-bounded infinitesimally symplectic operator is defined, strictly including the usual notion of positive self-adjoint extension in a complex Hilbert space; all such skew-adjoint extensions are parametrized, as in the von Neumann or Birman-Krein-Vishik theories. Finally, the unique complex Hilbertian structure—formulated on the space of solutions of the covariant Klein-Gordon equation in generic conformal perturbations of flat space—is uniquely determined by invariance under the scattering operator. The invariant Hilbert structure is explicitly calculated to first order for an infinite-dimensional class of purely time-dependent metric perturbations, and higher-order contributions are rigorously estimated.     

Homogeneous quantization and multiplicities of group representations

Let B be a compact manifold. A cone over B is a principal R⁺-bundle, X, with base B. Let (a, x) → ?_a(x) be the mapping associated with the action of a? R⁺ on X. X is called a symplectic cone if it possesses a symplectic form, ω, such that $\text{?}{}_{\mathrm{a}}{}^1\text{ω = aω}$. A compact Lie group, G, is said to act in a homogeneous fashion on X if it acts on X in such a way that both ω and the principal bundle structure are preserved. It is known that to such an action one can associate in a fairly canonical way a representation of G on a Hilbert space H. (See [3].) In this paper we propose a symplectic recipe for the multiplicities with which H decomposes into G-irreducibles and show that this recipe is correct “generically”.     

Symplectic resolutions for nilpotent orbits (II)

Based on our previous work, Fu (Invent. Math. 151 (2003) 167–186), we prove that, given any two projective symplectic resolutions Z₁ and Z₂ of a nilpotent orbit closure in a complex simple Lie algebra of classical type, Z₁ is deformation equivalent to Z₂. This provides support for a ‘folklore’ conjecture on symplectic resolutions for symplectic singularities. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The symplectic reduced spaces of a Poisson actionLes espaces réduits symplectiques d'une action de Poisson

During the last thirty years, symplectic or Marsden–Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this Note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. In other words, the right category to obtain symplectically reduced spaces is that of Poisson manifolds acted upon canonically by a Lie group. To cite this article: J.-P. Ortega, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 999–1004.     

Spectral flexibility of symplectic manifolds <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">M</Emphasis>

We consider Riemannian metrics compatible with the natural symplectic structure on T ² × M, where T ² is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ₁. We show that λ₁ can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic question.     

On Odd Symplectic Schur Functions

Proctor defined combinatorially a family of Laurent Polynomials, called odd symplectic Schur functions, indexed by pairs (λ, c), where λ is partition andcis a column length of λ. A conjecture of Proctor (Invent. Math.92,1988, 307–332) includes the statement that the odd symplectic Schur functions are actually characters ofSp(2n + 1, C). The purpose of the present note is to prove this.     

Global Weak Solutions to the Compressible Navier-Stokes Equations in the Exterior Domain with Spherically Symmetric Data

In this paper, we prove the global existence of weak solutions to the full compressible Navier-Stokes equations in the domain exterior to a ball in R ⁿ (n=2,3) and with spherically symmetric data.     

Graded generalizations of Weyl- and Clifford algebras

Graded skew bilinear forms {,} on graded vector spaces V are defined such that their restrictions to the even resp. odd subspaces are skew resp. odd. Over such graded symplectic vector spaces a (universal) factor algebra of the tensor algebra of V is described which reduces to a Weyl- resp. Clifford algebra if only one even resp. odd subspace is nontrivial. Introducing the total graduation on this polynomial algebra and graded symmetrization it is shown that the elements up to second power are closed under graded commutation. If the graduation is of type Z₂ the elements of second power are a Lie-graded algebra and this is the only graduation for which this is true. The graded commutation relations of this algebra are calculated. It is isomorphic to the graded symplectic algebra of (V,{,}) which is contained in the graded derivation algebra of the graded Heisenberg algebra of elements up to first power.