82 research outputs found
Lectures on the free period Lagrangian action functional
In this expository article we study the question of the existence of periodic
orbits of prescribed energy for classical Hamiltonian systems on compact
configuration spaces. We use a variational approach, by studying how the
behavior of the free period Lagrangian action functional changes when the
energy crosses certain values, known as the Ma\~n\'e critical values.Comment: This expository article is the outcome of two series of lectures that
the author gave at two summer schools at the Korea Institute for Advanced
Study of Seul in 2010 and at the Universit\'e de Neuch\^atel in 201
A Morse complex for Lorentzian geodesics
We prove the Morse relations for the set of all geodesics connecting two
non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We
overcome the difficulties coming from the fact that the Morse index of every
geodesic is infinite, and from the lack of the Palais-Smale condition, by using
the Morse complex approach.Comment: 19 pages; updated references, final versio
When the Morse index is infinite
Let f be a smooth Morse function on an infinite dimensional separable Hilbert
manifold, all of whose critical points have infinite Morse index and co-index.
For any critical point x choose an integer a(x) arbitrarily. Then there exists
a Riemannian structure on M such that the corresponding gradient flow of f has
the following property: for any pair of critical points x,y, the unstable
manifold of x and the stable manifold of y have a transverse intersection of
dimension a(x)-a(y).Comment: 10 page
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 Gromov's
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.Comment: Final version, identical to the published one. Added comments about
the relationship with a conjecture of Viterbo and related references. Some of
the results of this paper are contained in our previous preprint
arXiv:1105.2931, which is no longer updated and will never become a published
articl
On the global stable manifold
We give an alternative proof of the stable manifold theorem as an application
of the (right and left) inverse mapping theorem on a space of sequences. We
investigate the diffeomorphism class of the global stable manifold, a problem
which in the general Banach setting gives rise to subtle questions about the
possibility of extending germs of diffeomorphisms.Comment: 18 pages, LaTeX2e file. Revised version (corrected typos,
introduction slightly extended). To appear in Studia Mathematic
Estimates and computations in Rabinowitz-Floer homology
The Rabinowitz-Floer homology of a Liouville domain W is the Floer homology
of the free period Hamiltonian action functional associated to a Hamiltonian
whose zero energy level is the boundary of W. It has been introduced by K.
Cieliebak and U. Frauenfelder. Together with A. Oancea, the same authors have
recently computed the Rabinowitz-Floer homology of the cotangent disk bundle
D^*M of a closed manifold M, by establishing a long exact sequence. The first
aim of this paper is to present a chain level construction of this exact
sequence. In fact, we show that this sequence is the long homology sequence
induced by a short exact sequence of chain complexes, which involves the Morse
chain complex and the Morse differential complex of the energy functional for
closed geodesics on M. These chain maps are defined by considering spaces of
solutions of the Rabinowitz-Floer equation on half-cylinders, with suitable
boundary conditions which couple them with the negative gradient flow of the
geodesic energy functional. The second aim is to generalize this construction
to the case of a fiberwise uniformly convex compact subset W of T^*M whose
interior part contains a Lagrangian graph. Equivalently, W is the energy
sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any
energy level which is larger than the strict Mane' critical value of L. In this
case, the energy functional for closed geodesics is replaced by the free period
Lagrangian action functional associated to a suitable calibration of L. An
important issue in our analysis is to extend the uniform estimates for the
solutions of the Rabinowitz-Floer equation to Hamiltonians which have quadratic
growth in the momenta. These uniform estimates are obtained by suitable
versions of the Aleksandrov maximum principle.Comment: Revised versio
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 some examples of
symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which
would be impossible in finite dimensions.Comment: Added application to the nonlinear Schr\"odinger equatio
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