Alberto Abbondandolo

A smooth pseudo-gradient for the Lagrangian action functional

Abstract We study the action functional associated to a smooth Lagrangian function on the tangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is suffcient to associate a Morse complex to the Lagrangian action functional.

A Morse complex for infinite dimensional manifolds, Part I

Abstract In this paper and in the forthcoming Part II, we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M , possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle—or more generally an essential subbundle—of the tangent bundle of M , suitably related with the gradient flow of f . This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y : finite dimensionality of W , possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W , classification, up to topological conjugacy, of the gradient flow on the closure of W , in the case dim W = 2 .

LECTURES ON THE MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

After reviewing some classical results about hyperbolic dynamics in a Banach setting, we describe the Morse complex for gradient-like flows on an infinite-dimensional Banach manifold M, under the assumption that rest points have finite Morse index. Then we extend these ideas to rest points with infinite Morse index and co-index, by using a suitable subbundle of the tangent bundle of M as a comparison object.

A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces

A generalized cohomology, similar to Szulkins cohomology but with more general functorial properties, is constructed. This theory is used to define a relative Morse index and to prove relative Morse relations for strongly indefinite functionals on Hilbert spaces.

HOW LARGE IS THE SHADOW OF A SYMPLECTIC BALL

Consider the image of a 2n-dimensional unit ball by an open symplectic embedding into the standard symplectic vector space of dimension 2n. Its 2k-dimensional shadow is its orthogonal projection into a complex subspace of real dimension 2k. Is it true that the volume of this 2k-dimensional shadow is at least the volume of the unit 2k-dimensional ball? This statement is trivially true when k = n, and when k = 1 it is a reformulation of Gromovs non-squeezing theorem. Therefore, this question can be considered as a middle-dimensional generalization of the non-squeezing theorem. We investigate the validity of this statement in the linear, nonlinear and perturbative setting.

A non-squeezing theorem for convex symplectic images of the Hilbert ball

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss the role of infinite-dimensional non-squeezing results in the study of Hamiltonian PDEs and show some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

Given the cotangent bundle T ∗ Q of a smooth manifold with its canonical symplectic structure, and a Hamiltonian function on T ∗ Q which is fiberwise asymptot- ically quadratic, its well-defined Floer homology with the pair-of-pants ring structure is ring-isomorphic to the singular homology of the free loop space of Q endowed with its loop product. The analogous statement is true for the based loop space versions and the Pontrjagin product. This article gives an overview of the construction of this ring isomorphism which is based on Legendre duality and moduli spaces of flow trajectories of hybrid type, which are half Floer trajectories for the Hamiltonian problem and half Morse trajectories for the Lagrangian problem.

On the Morse index of Lagrangian systems

We give a new proof of the identity between the Morse index of a periodic solution of a Lagrangian system, positive definite in the velocities, and its Maslov index. Furthermore, we give an interpretation of the Maslov index when the Lagrangian system is not positive definite in the velocities. Our approach consists in relating the Morse index of the solution to a relative Morse index obtained passing to the Hamiltonian formulation. The main concepts used are the notion of commensurable subspaces, relative dimension, Fredholm pairs and relative index of strongly indefinite bilinear forms.

The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

Abstract We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range ( e 0 , e 1 )

When the Morse index is infinite

{(e_{0},e_{1})}