Publications (39)19.17 Total impact

Article: Symplectic Tate homology
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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.05/2014; 
Article: Symplectic Tate homology
04/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We give in this article necessary and sufficient conditions on the topology of rationally and polynomially convex domains.05/2013;  [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.05/2013; 
Article: Flexible Weinstein manifolds
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ABSTRACT: This survey on flexible Weinstein manifolds is, essentially, an extract from our recent joint book.05/2013;  [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 twosphere. 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.10/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We prove that every stable Hamiltonian structure on a closed oriented threemanifold is stably homotopic to one which is supported (with suitable signs) by an open book. Comment: 30 pages, 6 figures12/2010;  [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 MorseBott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on the 3sphere is homotopic to a stable Hamiltonian structure which cannot be embedded in 4space. 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 figures03/2010;  [Show abstract] [Hide abstract]
ABSTRACT: This paper studies dynamical systems in the twisted cotangent bundle of a Riemannian manifold (M,g). The twisted cotangent bundle of M is T * M with the symplectic form dθ+τ * σ where θ is the canonical form (equals to pdq in conjugate coordinates), τ:T * M→M is the canonical projection and σ is a closed 2form on M vanishing on π 2 (M). The authors consider particular Hamiltonian systems given by Hamiltonian functions H(λ)=1 2g(λ,λ)+U(τ(λ)) where U:M→ℝ is a potential. Physically, σ plays the role of a magnetic field on M. The main focus of the paper is on the change of the symplectic properties of the energy level sets H 1 (k) when k is below, equal or above a particular value of the energy called Mañé’s critical value defined by c:=c(g,σ,U)=inf dθ=π * σ sup q∈M ˜ H∘π ˜(θ q ), where π:M ˜→M is the universal cover and π ˜ is its lift to T * M. From the dynamical point of the view the interest is on whether the Hamiltonian flow on H 1 (k) admits a closed orbit in each non trivial free homotopy class of loops (similar to a geodesic flow) or a contractible periodic orbit. The authors are also interested in the displaceability of H 1 (k) and on whether or not H 1 (k) is stable (as defined in [H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics. Birkhäuser Advanced Texts. Basel: Birkhäuser (1994; Zbl 0805.58003)]) or virtually contact (i.e., of contact type when lifted to a finite cover). The authors give evidence of the following paradigms in several cases: If k<c, then H 1 (k) is displaceable and stable. If k=c, then H 1 (k) is nondisplaceable but nonstable (hence noncontact). If k>c, then H 1 (k) is nondisplaceable, virtually contact and the Hamiltonian flow has a closed orbit in every non trivial free homotopy class of loops. The tool used to study these properties is Rabinovitz Floer homology (RFH). It is an invariant introduced in [K. Cieliebak and U. A. Frauenfelder, Pac. J. Math. 239, No. 2, 251–316 (2009; Zbl 1221.53112)] where it was defined for hypersurfaces of restricted contact type. Roughly speaking, for a hypersurface Σ described as the 0 level set of a Hamiltonian H ¯ in a geometrically bounded symplectic manifold (V,ω), RFH is a Floer type theory for the functional defined on ℒ×ℝ where ℒ is the set of contractible loops in V by A(v,η)=∫ D 2 v ¯ * ωη∫ 0 1 H ¯(v(t))dt where v ¯ is a disk with boundary v. Here, η plays the role of a Lagrange multiplier for the standard Floer action functional, constraining the critical points to loops on Σ. In the paper under review, a thorough study of Rabinovitz Floer homology is done in the more general case of stable tame hypersurfaces in geometrically bounded symplectic manifolds and it is shown that it is well defined, i.e., it is invariant under deformation of these surfaces. In this case the compactness result differs from previous work. It is also shown that for the virtually contact case when π 1 (Σ) injects in π 1 (M) then RFH(Σ) is defined as a straightforward generalisation of [K. Cieliebak and U. A. Frauenfelder, Pac. J. Math. 239, No. 2, 251–316 (2009; Zbl 1221.53112)]. It is proved that if Σ is displaceable then RFH(Σ) vanishes, and when RFH(Σ) vanishes then it carries a periodic orbit contractible in V. This allows to prove existence of a closed characteristic on some energy level on H 1 (k) when k<c on the one hand and non displaceabilty of H 1 (k) when k>c on the other in several particular cases. Finally, the paper contains also some explicit computations for such systems in the case of some homogeneous spaces. Those computation alsos provide evidences (and proofs in some cases) of the previous paradigms.Geometry & Topology 01/2010; 14(3):17651870. · 0.97 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 pagesJournal of the Korean Mathematical Society 11/2009; · 0.32 Impact Factor 
Article: Stability is not open
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ABSTRACT: We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable. Comment: 9 pages08/2009;  [Show abstract] [Hide abstract]
ABSTRACT: We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean selfintersection. As a consequence, we show that the number of intersections of such holomorphic curves with the selfintersection 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 RabinowitzFloer 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 RabinowitzFloer 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.04/2009;  [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 nonzero 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 1/4pinched negative curvature and nonexact magnetic fields all sufficiently high energy levels are nonstable, 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 Topology03/2009;  [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.11/2007;  [Show abstract] [Hide abstract]
ABSTRACT: We show that a nondegenerate tight contact form on the 3sphere 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 ConleyZehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4space.08/2007;  [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.07/2007;  [Show abstract] [Hide abstract]
ABSTRACT: We use Donaldson hypersurfaces to construct pseudocycles which define GromovWitten 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 
Article: Quantitative symplectic geometry
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ABSTRACT: While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the socalled 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.07/2005;  [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
Publication Stats
647  Citations  
19.17  Total Impact Points  
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Institutions

2002–2003

LudwigMaximilianUniversity of Munich
 Mathematisches Institut
München, Bavaria, Germany


2001–2002

Stanford University
 Department of Mathematics
Stanford, CA, United States


1997

ETH Zurich
Zürich, Zurich, Switzerland
