Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

Given a symplectic surface [Formula: see text] of genus [Formula: see text], we show that the free group with two generators embeds into every asymptotic cone of [Formula: see text], where [Formula: see text] is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds.

Symplectic capacities

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

On the existence of a Hofer type metric for Poisson manifolds

An analogue of the Hofer metric [Formula: see text] on the Hamiltonian group [Formula: see text] of a Poisson manifold [Formula: see text] can be defined, but there is the problem of its nondegeneracy. First, we observe that [Formula: see text] is a genuine metric on [Formula: see text], when the union of all proper leaves of the corresponding symplectic foliation is dense. Next, we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable, if it can be realized as the space of units of a symplectic groupoid. Our main result states that [Formula: see text] is a Hofer type metric for every Poisson manifold, which admits a Hausdorff integration.

DESCENT AND C0-RIGIDITY OF SPECTRAL INVARIANTS ON MONOTONE SYMPLECTIC MANIFOLDS

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend from [Formula: see text] to Hamc(M\U). Furthermore, we show that these invariants are continuous with respect to the C0-topology on Hamc(M\U).We apply the above results to Hofer geometry and establish unboundedness of the Hofer diameter of Hamc(M\U) for stably displaceable U. We also answer a question of F. Le Roux about C0-continuity properties of the Hofer metric.

CALABI QUASIMORPHISMS FOR THE SYMPLECTIC BALL

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show that the (Lagrangian) Clifford torus in the complex projective space cannot be displaced from itself by a Hamiltonian isotopy.