[show abstract][hide abstract] ABSTRACT: We investigate the Cartan and Finsler geometry of the rotating Kepler
problem, a limit case of the restricted three body problem that arises if the
mass of the one of the primaries goes to zero. We show that the Hamiltonian for
the rotating Kepler problem can be regarded as the Legendre transform of a
certain family of Finsler metrics on the two-sphere. For very negative energy
levels, these Finsler metrics are close to the round metric, and the associated
flag curvature is hence positive. On the other hand, we show that the flag
curvature can become negative once the energy level becomes sufficiently high.
[show abstract][hide abstract] ABSTRACT: We prove that every stable Hamiltonian structure on a closed oriented three-manifold is stably homotopic to one which is supported (with suitable signs) by an open book. Comment: 30 pages, 6 figures
[show abstract][hide abstract] ABSTRACT: In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; there exist stable Hamiltonian structures that are not homotopic to a positive contact structure; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on the 3-sphere is homotopic to a stable Hamiltonian structure which cannot be embedded in 4-space. Moreover, we derive a structure theorem in dimension three and classify stable Hamiltonian structures supported by an open book. We also discuss implications for the foundations of symplectic field theory. Comment: 101 pages, 3 figures
[show abstract][hide abstract] ABSTRACT: Given a Morse function on a manifold whose moduli spaces of gradient flow lines for each action window are compact up to breaking one gets a bidirect system of chain complexes. There are different possibilities to take limits of such a bidirect system. We discuss in this note the relation between these different limits. Comment: 26 pages
[show abstract][hide abstract] ABSTRACT: We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean self-intersection. As a consequence, we show that the number of intersections of such holomorphic curves with the self-intersection locus is uniformly bounded in terms of the Hofer energy.
Journal of Symplectic Geometry 04/2009; · 0.97 Impact Factor
[show abstract][hide abstract] ABSTRACT: The Rabinowitz-Floer homology groups $RFH_*(M,W)$ are associated to an exact embedding of a contact manifold $(M,\xi)$ into a symplectic manifold $(W,\omega)$. They depend only on the bounded component $V$ of $W\setminus M$. We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$, which in turn maps to Rabinowitz-Floer homology $RFH_*(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RFH_*(ST^*L,T^*L)$, where $ST^*L$ is the unit cosphere bundle of a closed manifold $L$. As an application, we prove that the image of an exact contact embedding of $ST^*L$ (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided $\dim L\ge 4$ and the embedding induces an injection on $\pi_1$. In particular, $ST^*L$ does not admit an exact contact embedding into a subcritical Stein manifold if $L$ is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings.
[show abstract][hide abstract] ABSTRACT: We study the dynamics and symplectic topology of energy hypersurfaces of mechanical Hamiltonians on twisted cotangent bundles. We pay particular attention to periodic orbits, displaceability, stability and the contact type property, and the changes that occur at the Mane critical value c. Our main tool is Rabinowitz Floer homology. We show that it is defined for hypersurfaces that are either stable tame or virtually contact, and it is invariant under under homotopies in these classes. If the configuration space admits a metric of negative curvature, then Rabinowitz Floer homology does not vanish for energy levels k>c and, as a consequence, these level sets are not displaceable. We provide a large class of examples in which Rabinowitz Floer homology is non-zero for energy levels k>c but vanishes for k<c, so levels above and below c cannot be connected by a stable tame homotopy. Moreover, we show that for strictly 1/4-pinched negative curvature and non-exact magnetic fields all sufficiently high energy levels are non-stable, provided that the dimension of the base manifold is even and different from two. Comment: 101 pages, no figures, final version appeared in Geometry and Topology
[show abstract][hide abstract] ABSTRACT: In this paper we construct the Floer homology for an action functional which was introduced by Rabinowitz and prove a vanishing theorem. As an application, we show that there are no displaceable exact contact embeddings of the unit cotangent bundle of a sphere of dimension greater than three into a convex exact symplectic manifold with vanishing first Chern class. This generalizes Gromov's result that there are no exact Lagrangian embeddings of a sphere into a complex vector space.
[show abstract][hide abstract] ABSTRACT: We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.
[show abstract][hide abstract] ABSTRACT: We outline a program for incorporating holomorphic curves with Lagrangian boundary conditions into symplectic field theory, with an emphasis on ideas, geometric intuition, and a description of the resulting algebraic structures.
[show abstract][hide abstract] ABSTRACT: We use Donaldson hypersurfaces to construct pseudo-cycles which define Gromov-Witten invariants for any symplectic manifold which agree with the invariants in the cases where transversality could be achieved by perturbing the almost complex structure.
Journal of Symplectic Geometry 03/2007; · 0.97 Impact Factor
[show abstract][hide abstract] ABSTRACT: While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. In this paper we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. We also describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion we mention many open problems.
[show abstract][hide abstract] ABSTRACT: Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder recently proved a general compactness result for moduli spaces of punctured holomorphic curves arising in symplectic field theory. In this paper we present an alternative proof of this result. The main idea is to determine a priori the levels at which holomorphic curves split, thus reducing the proof to two separate cases: long cylinders of small area, and regions with compact image. The second case requires a generalization of Gromov compactness for holomorphic curves with free boundary.
Journal of Symplectic Geometry 01/2005; · 0.97 Impact Factor
[show abstract][hide abstract] ABSTRACT: In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.
[show abstract][hide abstract] ABSTRACT: In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a K\"ahler-Einstein manifold. An immediate consequence is that in K\"ahler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.
Journal of Symplectic Geometry 11/2003; · 0.97 Impact Factor
[show abstract][hide abstract] ABSTRACT: In this paper we obtain new topological restrictions on Lagrangian embeddings into subcritical Stein manifolds. We also extend
previous results of Gromov, Oh, Polterovich and Viterbo on Lagrangian submanifolds of ℂ
to the more general case of subcritical Stein manifolds.
Israel Journal of Mathematics 11/2002; 127(1):221-244. · 0.65 Impact Factor
[show abstract][hide abstract] ABSTRACT: We show that a small neighborhood of a closed symplectic submanifold in a geometrically bounded aspherical symplectic manifold has non-vanishing symplectic homology. As a consequence, we establish the existence of contractible closed characteristics on any thickening of the boundary of the neighborhood. When applied to twisted geodesic flows on compact symplectically aspherical manifolds, this implies the existence of contractible periodic orbits for a dense set of low energy values.