Helmut Hofer

Introduction to Symplectic Field Theory

We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new invariants of contact manifolds and their Legendrian submanifolds. Moreover, we hope that the applications of SFT go far beyond this framework.1

Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three

l. In t roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 1.1. The Weinstein conjecture . . . . . . . . . . . . . . . . . . . . . . . . 515 1.2. The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 1.3. Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 2. Local F redho lm theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 2.1. Similar i ty principle and consequences . . . . . . . . . . . . . . . . . . . 525 2.2. N_ • M as a lmost complex manifold and local fillings . . . . . . . . . . . . . 529 2.3. Local propert ies of ho lomorphic curves . . . . . . . . . . . . . . . . . . 531 3. Bubbl ing off analysis and global Fredholm theory . . . . . . . . . . . . . . . . 533 3.l. Grad ien t bounds imply Ca -bounds . . . . . . . . . . . . . . . . . . . . 533 3.2. Bubbing off analysis of ho lomorphic planes . . . . . . . . . . . . . . . . . 534 3.3. Grad ien t bounds near the boundary . . . . . . . . . . . . . . . . . . . . 537 4. Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . 544 4.1. Set-up for a global Fredholm theory . . . . . . . . . . . . . . . . . . . . 544 4.2. Bishops theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 4.3. Proof of theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 4.4. Proof of theorems 8 and 11 . . . . . . . . . . . . . . . . . . . . . . . 547 4.5. Uniqueness results for foliations . . . . . . . . . . . . . . . . . . . . . 547 5. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 5.1. Per turbing surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 5.2. Ki l l ing crit ical points of hypersurfaces . . . . . . . . . . . . . . . . . . . 552 5.3. Adding a thin tube to a sphere . . . . . . . . . . . . . . . . . . . . . . 560

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).

On the topological properties of symplectic maps

In this paper we show that symplectic maps have surprising topological properties. In particular, we construct an interesting metric for the symplectic diffeomorphism groups, which is related, but not obviously, to the topological properties of symplectic maps and phase space geometry. We also prove a certain number of generalised symplectic fixed point theorems and give an application to a Hamiltonian system.

Floer homology and Novikov rings

We prove the Arnold conjecture for compact symplectic manifolds under the assumption that either the first Chern class of the tangent bundle vanishes over π2(M) or the minimal Chern number is at least half the dimension of the manifold. This includes the important class of Calabi-Yau manifolds. The key observation is that the Floer homology groups of the loop space form a module over Novikov’s ring of generalized Laurent series. The main difficulties to overcome are the presence of holomorphic spheres and the fact that the action functional is only well defined on the universal cover of the loop space with a possibly dense set of critical levels.

Transversality in elliptic Morse theory for the symplectic action

Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder. These results play a central role in the denition of symplectic Floer homology and hence in the proof of the Arnold conjecture. There is currently no other reference to these transversality results in the open literature. Our approach does not require Aronszajn’s theorem. Instead we derive the unique continuation theorem from a generalization of the Carleman similarity principle.

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.

On genericity for holomorphic curves in four-dimensional almost-complex manifolds

We consider spaces of immersed (pseudo-)holomorphic curves in an almost complex manifold of dimension four. We assume that they are either closed or compact with boundary in a fixed totally real surface, so that the equation for these curves is elliptic and has a Fredholm index. We prove that this equation is regular if the Chern class is ≥ 1 (in the case with boundary, if the ambient Maslov number is ≥ 1). Then the spaces of holomorphic curves considered will be manifolds of dimension equal to the index.

Convex Hamiltonian energy surfaces and their periodic trajectories

In this paper we introduce symplectic invariants for convex Hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations. In particular we show that they can be used to prove multiplicity results for the number of periodic trajectories.

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.