Publications (48)32.71 Total impact
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ABSTRACT: The conormal Lagrangian $L_K$ of a knot $K$ in $\mathbb{R}^3$ is the submanifold of the cotangent bundle $T^* \mathbb{R}^3$ consisting of covectors along $K$ that annihilate tangent vectors to $K$. By intersecting with the unit cotangent bundle $S^* \mathbb{R}^3$, one obtains the unit conormal $\Lambda_K$, and the Legendrian contact homology of $\Lambda_K$ is a knot invariant of $K$, known as knot contact homology. We define a version of string topology for strings in $\mathbb{R}^3 \cup L_K$ and prove that this is isomorphic in degree 0 to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree 0 knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in $T^* \mathbb{R}^3$ with boundary on $\mathbb{R}^3 \cup L_K$.  [Show abstract] [Hide abstract]
ABSTRACT: We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and explore some of its consequences.  [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 biLie 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. 
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.  [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. 
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 $\underrightarrow {H}\underleftarrow {T}(W;\mathbb {Q})$ has the fixed point property and is therefore isomorphic to $H(W;\mathbb {Q}) \otimes _\mathbb {Q} \mathbb {Q}[u,u^{1}]$, 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: This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3manifold. 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 3manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis. 
Article: Flexible Weinstein manifolds
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ABSTRACT: This survey on flexible Weinstein manifolds is, essentially, an extract from our recent joint book.  [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 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.  [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 Wassermantype theorem for the Rabinowitz action functional”, where we showed that the Rabinowitz action functional is generically MorseBott and the MorseBott 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 threemanifold is stably homotopic to one which is supported (with suitable signs) by an open book.  [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 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  [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 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 
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 pages  [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.  [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.
Publication Stats
929  Citations  
32.71  Total Impact Points  
Top Journals
Institutions

20102014

Universität Augsburg
 Institute of Mathematics
Augsberg, Bavaria, Germany


20052011

Technische Universität München
 Department of Mathematical Statistics
München, Bavaria, Germany


20022010

LudwigMaximiliansUniversity of Munich
 Mathematisches Institut
München, Bavaria, Germany


20002002

Stanford University
 Department of Mathematics
Stanford, CA, United States


1998

Harvard University
 Department of Mathematics
Cambridge, MA, United States


1997

ETH Zurich
Zürich, Zurich, Switzerland
