Kai Cieliebak

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

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Publications (42)28.16 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;
  • Publicationes mathematicae 05/2014; 84(3). · 0.32 Impact Factor
  • 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.
    Inventiones mathematicae 05/2013; · 2.26 Impact Factor
  • 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;
  • Kai Cieliebak, Urs Adrian Frauenfelder
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    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.
    Pacific Journal of Mathematics 01/2011; 249(2). · 0.47 Impact Factor
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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
    Journal of Topology 12/2010; · 0.67 Impact Factor
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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;
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    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 2-form 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 non-displaceable but non-stable (hence non-contact). If k>c, then H -1 (k) is non-displaceable, 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 hyper-surface 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 ¯ * ω-η∫ 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):1765-1870. · 0.97 Impact Factor
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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
    Annales- Institut Fourier 08/2009; · 0.53 Impact Factor
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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.
    Annales Scientifiques de l École Normale Supérieure 04/2009; · 0.96 Impact Factor
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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.
    Pacific Journal of Mathematics 11/2007; · 0.47 Impact Factor
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    S. Baader, K. Cieliebak, T. Vogel
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    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; · 0.40 Impact Factor
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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.
    Journal of Modern Dynamics 08/2007; · 0.94 Impact Factor
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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;

Publication Stats

665 Citations
28.16 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