Eduard Zehnder

The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold

Abstract : The following conjecture of V. I. Arnold is proved: every measure preserving diffeomorphism of the torus T2, which is homologeous to the identity, and which leaves the center of mass invariant, possesses at least 3 fixed points. The proof of this global fixed point theorem does not make use of the generating function technique. The theorem is a consequence of the statement that a Hamiltonian vector-field on a torus T2n, which depends periodically on time, possesses at least (2n+1) forced oscillations. These periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows. A second conjecture of V. I. Arnold proved concerns a Birkhoff-Lewis type fixed point theorem for symplectic maps. Additional keywords; Periodic functions; Variational principles; Equations; Reprints. (Author)

Compactness results in Symplectic Field Theory

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in (4). We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromovs compactness theorem in (8) as well as compactness theorems in Floer homology theory, (6, 7), and in contact geometry, (9, 19).

The dynamics on three-dimensional strictly convex energy surfaces

We show that a Hamiltonian flow on a three-dimensional strictly convex energy surface S C R4 possesses a global surface of section of disc type. It follows, in particular, that the number of its periodic orbits is either 2 or oc, by a recent result of J. Franks on area-preserving homeomorphisms of an open annulus in the plane. The construction of this surface of section is based on partial differential equations of Cauchy-Riemann type for maps from punctured Riemann surfaces into R x S3 equipped with special almost complex structures.

KAM theory in configuration space

A new approach to the Kolmogorov-Arnold-Moser theory concerning the existence of invariant tori having prescribed frequencies is presented. It is based on the Lagrangian formalism in configuration space instead of the Hamiltonian formalism in phase space used in earlier approaches. In particular, the construction of the invariant tori avoids the composition of infinitely many coordinate transformations. The regularity results obtained are applied to invariant curves of monotone twist maps. The Lagrangian approach has been prompted by a recent study of minimal foliations for variational problems on a torus by J. Moser.

Periodic Solutions of Asymptotically Linear Hamiltonian Systems.

We prove existence and multiplicity results for periodic solutions of time dependent and time independent Hamiltonian equations, which are assumed to be asymptotically linear. The periodic solutions are found as critical points of a variational problem in a real Hilbert space. By means of a saddle point reduction this problem is reduced to the problem of finding critical points of a function defined on a finite dimensional subspace. The critical points are then found using generalized Morse theory and minimax arguments.

Properties of pseudoholomorphic curves in symplectisations I: Asymptotics

Abstract Given an oriented, compact, 3-dimenional contact manifold (M, λ) we study maps u ˜ = ( a , u ) : ℂ → ℝ × M satisfying the Cauch-Riemann type equation u ˜ s + J ˜ ( u ˜ ) u ˜ t = 0 , with a very special almost complex structure J ˜ related to the contact form λ on M. If the energy is positive and bounded, 0 E ( ( u ˜ ) ∞ , then the asymptotic behavior of u : ℂ → M as |z| → ∞ is intimately related to the dynamics of the Reeb vector field Xλ on M. Assuming the periodic solutions of Xλ to be non degenerate, we shall show that lim R → ∞ u ( R e 2 π i t ) = x ( T t ) for a T-periodic solution x with E ( u ˜ ) = T . The main result is an asymptotic formula which demonstrates the exponential nature of this limit. Some consequences for the geometry of the maps u : ℂ → M are deduced.

Properties of pseudo-holomorphic curves in symplectisations II: Embedding controls and algebraic invariants

In the following we look for conditions on a finite energy plane ũ : = (a, u) : ℂ → ℝ × M, which allow us to conclude that the projection into the manifold M, u : ℂ → M, is an embedding. For this purpose we shall introduce several algebraic invariants. Finite energy planes have been introduced in [H] for the solution of A. Weinstein’s conjecture about closed characteristics on three dimensional contact manifolds. In order to recall the concept, we first start with some definitions from contact geometry.

A general Fredholm theory I: A splicing-based differential geometry

This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. These spaces, in general, are locally not homeomorphic to open sets in Banach spaces. The present paper describes some of the differential geometry of this new class of spaces. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory, and Symplectic Field Theory

A New Capacity for Symplectic Manifolds

Publisher Summary This chapter discusses the axioms for a symplectic capacity. At present, very little is known about the nature of a symplectic map. The axioms for a symplectic capacity are useful for a more systematic study of the symplectic embedding problem; they led to a new rigidity result. The axioms such as monotonicity, conformality, local nontriviality, and nontriviality do not determine a capacity function uniquely. There are many ways to construct different capacity functions. The capacity of every symplectic manifold is positive or ∞. Every capacity singles out the subgroup of homeomorphisms of R2n preserving the capacity. The elements of this distinguished group of homeomorphisms have the additional property that they are symplectic or anti-symplectic in case they are differentiable. The associated pseudogroup can be used to define a topological symplectic manifold.

Properties of Pseudoholomorphic Curves in Symplectizations III: Fredholm Theory

We shall study smooth maps ũ: S → ℝ x M of finite energy defined on the punctured Riemann surface S = S\Γ and satisfying a Cauchy-Riemann type equation Tũ ∘ j = Jũ ∘ Tũ for special almost complex structures J, related to contact forms A on the compact three manifold M. Neither the domain nor the target space are compact. This difficulty leads to an asymptotic analysis near the punctures. A Fredholm theory determines the dimension of the solution space in terms of the asymptotic data defined by non-degenerate periodic solutions of the Reeb vector field associated with λ on M, the Euler characteristic of S, and the number of punctures. Furthermore, some transversality results are established.