Kai Cieliebak

Ludwig-Maximilian-University of Munich, München, Bavaria, Germany

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Publications (38)18.2 Total impact

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    Peter Albers, Kai Cieliebak, Urs Frauenfelder
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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;
  • Peter Albers, Kai Cieliebak, Urs Frauenfelder
    04/2014;
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    Kai Cieliebak, Yakov Eliashberg
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    ABSTRACT: We give in this article necessary and sufficient conditions on the topology of rationally and polynomially convex domains.
    05/2013;
  • Kai Cieliebak, Yakov Eliashberg
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    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;
  • Kai Cieliebak, Yasha Eliashberg
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    ABSTRACT: This survey on flexible Weinstein manifolds is, essentially, an extract from our recent joint book.
    05/2013;
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    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.
    10/2011;
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    Kai Cieliebak, Evgeny Volkov
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    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
    12/2010;
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    Kai Cieliebak, Evgeny Volkov
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    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
    03/2010;
  • Geometry & Topology - GEOM TOPOL. 01/2010; 14(3):1765-1870.
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    Kai Cieliebak, Urs Frauenfelder
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    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; · 0.32 Impact Factor
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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 pages
    08/2009;
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    Kai Cieliebak, Tobias Ekholm, Janko Latschev
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    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
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    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.
    04/2009;
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    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
    03/2009;
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    Kai Cieliebak, Urs Frauenfelder
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    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;
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    F. Bourgeois, K. Cieliebak, T. Ekholm
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    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.
    08/2007;
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    Kai Cieliebak, Janko Latschev
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    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;
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    Kai Cieliebak, Klaus Mohnke
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    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
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    K. Cieliebak, H. Hofer, J. Latschev, F. Schlenk
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    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.
    07/2005;
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    K. Cieliebak, K. Mohnke
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    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

641 Citations
18.20 Total Impact Points

Institutions

  • 2002–2003
    • Ludwig-Maximilian-University of Munich
      • Mathematisches Institut
      München, Bavaria, Germany
  • 2001–2002
    • Stanford University
      • Department of Mathematics
      Stanford, CA, United States
  • 1997
    • ETH Zurich
      Zürich, Zurich, Switzerland