[Show abstract][Hide abstract] ABSTRACT: In this article we describe an algebraic framework which can be used in three
related but different contexts: string topology, symplectic field theory, and
Lagrangian Floer theory of higher genus. It turns out that the relevant
algebraic structure for all three contexts is a homotopy version of involutive
bi-Lie algebras, which we call IBL$_\infty$-algebras,
[Show abstract][Hide abstract] ABSTRACT: We use a neck stretching argument for holomorphic curves to produce
symplectic disks of small area and Maslov class with boundary on Lagrangian
submanifolds of nonpositive curvature. Applications include the proof of
Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic
space, the construction of a new symplectic capacity, obstructions to
Lagrangian embeddings into uniruled symplectic manifolds, a quantitative
version of Arnold's chord conjecture, and estimates on the size of Weinstein
neighbourhoods. The main technical ingredient is transversality for the
relevant moduli spaces of punctured holomorphic curves with tangency
conditions.
[Show abstract][Hide abstract] ABSTRACT: For a Liouville domain $W$ satisfying $c_1(W)=0$, we propose in this note two
versions of symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$
and $\underleftarrow{H}\underrightarrow{T}(W)$ which are related by a canonical
map $\kappa \colon \underrightarrow{H}\underleftarrow{T}(W) \to
\underleftarrow{H}\underrightarrow{T}(W)$. Our geometric approach to Tate
homology uses the moduli space of finite energy gradient flow lines of the
Rabinowitz action functional for a circle in the complex plane as a classifying
space for $S^1$-equivariant Tate homology. For rational coefficients the
symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$ has the
fixed point property and is therefore isomorphic to $H(W;\mathbb{Q}) \otimes
\mathrm{Q}[u,u^{-1}]$, where $\mathbb{Q}[u,u^{-1}]$ is the ring of Laurent
polynomials over the rationals. Using a deep theorem of Goodwillie, we
construct examples of Liouville domains where the canonical map $\kappa$ is not
surjective and examples where it is not injective.
[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: This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.
Ergodic Theory and Dynamical Systems 02/2014; DOI:10.1017/etds.2015.50 · 0.78 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This survey on the topology of Stein manifolds is an extract from our recent
joint book. It is compiled from two short lecture series given by the first
author in 2012 at the Institute for Advanced Study, Princeton, and the Alfred
Renyi Institute of Mathematics, Budapest.
[Show abstract][Hide abstract] ABSTRACT: We give in this article necessary and sufficient conditions on the topology
of rationally and polynomially convex domains.
[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: The paper in question [the authors, ibid. 239, No. 2, 251–316 (2009; Zbl 1221.53112)] included an appendix, titled “A Wasserman-type theorem for the Rabinowitz action functional”, where we showed that the Rabinowitz action functional is generically Morse-Bott and the Morse-Bott manifold is the disjoint union of the energy hypersurface itself, representing the constant Reeb orbits, and a circle for each Reeb orbit. The treatment of multiple covered Reeb orbits contained a gap, which is filled in this note.
[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.
Journal of Topology 12/2010; 7(3). DOI:10.1112/jtopol/jtt044 · 0.79 Impact Factor
[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 Mañé 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 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
[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
Journal of the European Mathematical Society 03/2010; 17(2). DOI:10.4171/JEMS/505 · 1.70 Impact Factor
[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
Journal of the Korean Mathematical Society 11/2009; 48(4). DOI:10.4134/JKMS.2011.48.4.749 · 0.51 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable. Comment: 9 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.
[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.
Annales Scientifiques de l École Normale Supérieure 04/2009; 43(6). · 1.52 Impact Factor
[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 null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-\chi(S)$. In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.
Journal of Knot Theory and Its Ramifications 09/2007; DOI:10.1142/S0218216509006999 · 0.41 Impact Factor