Leonid Polterovich

Calabi quasimorphism and quantum homology

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.

Rigid subsets of symplectic manifolds

We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Quasi-states and symplectic intersections

We establish a link between symplectic topology and a recently emerg\-ed branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures). In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.

Growth of maps, distortion in groups and symplectic geometry

Abstract.In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism:¶• The uniform norm of the differential of its n-th iteration;¶• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.¶We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups.

Partially ordered groups and geometry of contact transformations

Abstract. We prove that, for a class of contact manifolds, the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation number. Our approach is based on tools of Symplectic Topology.

Symplectic displacement energy for Lagrangian submanifolds

Recently H. Hofer defined a new symplectic invariant which has a beautiful dynamical meaning. In the present paper we study this invariant for Lagrangian submanifolds of symplectic manifolds. Our approach is based on Gromovs theory of pseudo-holomorphic curves.

Symplectic rigidity: Lagrangian submanifolds

This chapter is supposed to be a summary of what is known today about Lagrangian embeddings. We emphasise the difference between flexibility results, such as the h-principle of Gromov applied here to Lagrangian immersions (and also to the construction of examples of Lagrangian embeddings) and rigidity theorems, based on existence theorems for pseudo-holomorphic curves.

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 thes...

Sign and area in nodal geometry of Laplace eigenfunctions

The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.

Symplectic diffeomorphisms as isometries of Hofer's norm

Abstract In this paper, we extend the Hofer norm to the group of symplectic diffeomorphisms of a manifold. This group acts by conjugation on the group of Hamiltonian diffeomorphisms, so each symplectic diffeomorphism induces an isometry of the group Ham( M ) with respect to the Hofer norm. The C 0 -norm of this isometry, once restricted to the ball of radius α of Ham( M ) centered at the identity, gives a scale of norms r α on the group of symplectomorphisms. We conjecture that the subgroup of the symplectic diffeomorphisms which are isotopic to the identity and whose norms r α remain bounded when α → ∞ coincide with the group of Hamiltonian diffeomorphisms. We prove this conjecture for products of surfaces.