Nearby Lagrangians with vanishing Maslov class are homotopy equivalent

We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study an extension of the Fukaya category in which Lagrangians equipped with local systems of arbitrary dimension are admitted as objects, and prove that this extension is generated, in the appropriate sense, by local systems over a cotangent fibre. Whenever the cotangent bundle is simply connected, this generation statement is used to prove that every closed exact Lagrangian of vanishing Maslov index is simply connected. Finally, we borrow ideas from coarse geometry to develop a Fukaya category associated to the universal cover, allowing us to prove the result in the general case.     

Exact Lagrangian submanifolds in simply-connected cotangent bundles

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel–Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different approach.     

The fixed energy problem for a class of nonconvex singular Hamiltonian systems

We consider a noncompact hypersurface H in R2N which is the energy level of a singular Hamiltonian of “strong force” type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type.     

Functors and Computations in Floer Homology with Applications, I

This paper is concerned with Floer cohomology of manifolds with contact type boundary. In this case, there is no conjecture on this ring, as opposed to the compact case, where it is isomorphic to the usual cohomology (with the quantum product). We construct two mappings in Floer cohomology and prove some functorial properties of these two mappings. The first one is a map from the Floer cohomology of M to the relative cohomology of M modulo its boundary. The other is associated to a codimension zero embedding, and may be considered as a cohomological transfer. These maps are used to define some properties of symplectic manifolds with contact type boundary. These are algebraic versions of the Weinstein conjecture, asserting existence of closed characteristics on . This is proved for many cases, Euclidean space and subcritical Stein manifolds, vector bundles, products, cotangent bundles. It is also proved that the above property implies some restrictions on Lagrangian embeddings, and also yields in certain cases, existence results for holomorphic curves bounded by the Lagrange submanifold. The last section is devoted to applications of this existence result, to real forms of Stein manifolds and obstructions to polynomial convexity in Stein manifolds. Many of our applications rely on the computation of the Floer cohomology of a cotangent bundle, that is the subject of Part II. Submitted: December 1997, revised version: February 1999.     

Floer–Novikov homology and applications to Lagrangian submanifolds

We use a non-Hamiltonian version of Lagrangian Floer homology to prove that an exact Lagrangian submanifold in the cotangent bundle of the 3-torus T ³ must be diffeomorphic to T ³. This improves a previous result of Fukaya, Seidel and Smith.     

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.     

Hamiltonian systems with a discrete symmetry

If $\text{?: M → M}$ is an antisymplectic involution of a symplectic manifold M then the fixed set of ? is a Lagrangian submanifold L ? M. Moreover there exist cotangent bundle coordinates in a neighborhood of L in M such that ? in these coordinates maps a covector into its negative. Thus classical examples which have a discrete symmetry such as the restricted three-body problems are locally like a reversible system.     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

The Palais-Smale condition on contact type energy levels for convex Lagrangian systems

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Mañé's critical value of the lift of the Lagrangian to the universal cover, c _u (L), almost all energy levels have conjugate points. We in addition prove that if an energy level is of contact type, projects onto M and $M\ne{\mathbb T}^2We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma?é's critical value of the lift of the Lagrangian to the universal cover, c _u (L), almost all energy levels have conjugate points.We in addition prove that if an energy level is of contact type, projects onto M and , then the free time action functional of L+k satisfies the Palais-Smale condition.Partially supported by Conacyt, Mexico, grant 36496-E.     

Finiteness of $$\pi _1$$-sensitive Hofer–Zehnder capacity and equivariant loop space homology

We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer–Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.     

Torsions of Connections on Higher Order Cotangent Bundles

By a torsion of a general connection Γ on a fibered manifold Y → M we understand the Frölicher-Nijenhuis bracket of Γ and some canonical tangent valued one-form (affinor) on Y. Using all natural affinors on higher order cotangent bundles, we determine all torsions of general connections on such bundles. We present the geometrical interpretation and study some properties of the torsions.     

Well-posed infinite horizon variational problems on a compact manifold

We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i.e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian submanifold (well-projected to M) of the flow of extremals in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics.     

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non-Kähler) manifolds and in twistor spaces Z ⁴ⁿ⁺² over quaternionic Kähler manifolds Q ⁴ⁿ are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kähler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of M and the other factor is Lagrangian in the Kähler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight, and ten.     

Webs of Lagrangian tori in projective symplectic manifolds

For a Lagrangian torus A in a simply-connected projective symplectic manifold M, we prove that M has a hypersurface disjoint from a deformation of A. This implies that a Lagrangian torus in a compact hyperkähler manifold is a fiber of an almost holomorphic Lagrangian fibration, giving an affirmative answer to a question of Beauville’s. Our proof employs two different tools: the theory of action-angle variables for algebraically completely integrable Hamiltonian systems and Wielandt’s theory of subnormal subgroups.     

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M₃.     

Différentes estimations de supu×infu pour l'équation de la courbure scalaire prescrite en dimension n⩾3

On a Riemannian manifolds (M,g) of dimension n, we prove on compact set K⊂M, that the positive solutions of the equation of prescribed scalar curvature (and the equation of subcritical case) are uniformely bounded.In positive case, when the manifold is compact, we prove that sup_Mu×inf_Mu⩾c>0 if n⩾3 (respectively sup_Mu+inf_Mu⩾c is n=2).     

On the structure of conformally compact Einstein metrics

Let M be an (n + 1)-dimensional manifold with non-empty boundary, satisfying π ₁(M, ? M) = 0. The main result of this paper is that the space of conformally compact Einstein metrics on M is a smooth, infinite dimensional Banach manifold, provided it is non-empty. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with prescribed metric and stress–energy tensor at conformal infinity, again in dimension 4. This result also holds for Lorentzian–Einstein metrics with a positive cosmological constant.     

The basis monomial ring of a matroid

We define the basis monomial ring M_G of a matroid G and prove that it is Cohen-Macaulay for finite G. We then compute the Krull dimension of M_G, which is the rank over Q of the basis-point incidence matrix of G, and prove that dim B_G ≥ dim M_G under a certain hypothesis on coordinatizability of G, where B_G is the bracket ring of G.     

On the Floer homology of cotangent bundles

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L^∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W^1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.     

The Conley conjecture for the cotangent bundle

We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For the conservative systems, similar results have been proven by Lu and Mazzucchelli using convex Hamiltonians and Lagrangian methods. Our proof uses Floer homological methods from Ginzburg’s proof of the Conley conjecture for closed symplectically aspherical manifolds.