Liouville Domain Research Articles

Hamiltonian perturbations in contact Floer homology

We study the contact Floer homology \(\mathrm{HF}_*(W, h)\) introduced by Merry–Uljarević in [21], which associates a Floer-type homology theory with a Liouville domain W and a contact Hamiltonian h on its boundary. The main results investigate the behavior of \(\mathrm{HF}_*(W, h)\) under the perturbations of the input contact Hamiltonian h. In particular, we provide sufficient conditions that guarantee \(\mathrm{HF}_*(W, h)\) to be invariant under the perturbations. This can be regarded as a contact geometry analog of the continuation and bifurcation maps along the Hamiltonian perturbations of Hamiltonian Floer homology in symplectic geometry. As an application, we give an algebraic proof of a rigidity result concerning the positive loops of contactomorphisms for a wide class of contact manifolds.

On fillings of $$\partial (V\times {\mathbb {D}})$$

We show that any symplectically aspherical/Calabi-Yau filling of $Y:=\partial(V\times \mathbb{D})$ has vanishing symplectic cohomology for any Liouville domain $V$. In particular, we make no topological requirement on the filling and $c_1(V)$ can be nonzero. Moreover, we show that for any symplectically aspherical/Calabi-Yau filling $W$ of $Y$, the interior $\mathring{W}$ is diffeomorphic to the interior of $V\times \mathbb{D}$ if $\pi_1(Y)$ is abelian and $\dim V\ge 4$. And $W$ is diffeomorphic to $V\times \mathbb{D}$ if moreover the Whitehead group of $\pi_1(Y)$ is trivial.

Exotic symplectomorphisms and contact circle actions

Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard symplectic ball, [Formula: see text] implies a rather strict topological condition on the free contact circle actions on the standard contact sphere, [Formula: see text] We also prove an analogue for a Liouville domain and contact circle actions on its boundary. Applications include results concerning the symplectic mapping class group and the fundamental group of the group of contactomorphisms.

Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

AbstractThe aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex star-shaped hypersurfaces in ${\mathbb {R}}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.

Symplectic Banach–Mazur distances between subsets of ℂn

Following proposals of Ostrover and Polterovich, we introduce and study “coarse” and “fine” versions of a symplectic Banach–Mazur distance on certain open subsets of [Formula: see text] and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljević and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach–Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in [Formula: see text] admits quasi-isometric embeddings of [Formula: see text] for every finite dimension D. Our constructions are obtained from a general method of constructing [Formula: see text]-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given [Formula: see text]-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from [J. Gutt and M. Usher, Symplectically knotted codimension-zero embeddings between domains in [Formula: see text], Duke Math. J. 168 (2019) 2299–2363].

Positive topological entropy of positive contactomorphisms

A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $\Lambda$, and that $(M,\Lambda)$ is fillable by a Liouville domain $(W,\omega)$ with exact Lagrangian $L$ such that $\omega|_{\pi_2(W,L)}=0$. We show that if the exponential growth of the action filtered wrapped Floer homology of $(W,L)$ is positive, then every positive contactomorphism of $M$ has positive topological entropy. This result generalizes the result of Alves and Meiwes from Reeb flows to positive contactomorphisms, and it yields many examples of contact manifolds on which every positive contactomorphism has positive topological entropy, among them the exotic contact spheres found by Alves and Meiwes. A main step in the proof is to show that wrapped Floer homology is isomorphic to the positive part of Lagrangian Rabinowitz-Floer homology.

Morse\u2013Bott split symplectic homology

We associate a chain complex to a Liouville domain (overline{W}, mathrm{d}lambda ) whose boundary Y admits a Boothby–Wang contact form (i.e. is a prequantization space). The differential counts Floer cylinders with cascades in the completion W of overline{W}, in the spirit of Morse–Bott homology (Bourgeois in A Morse–Bott approach to contact homology, Ph.D. Thesis. ProQuest LLC, Stanford University, Ann Arbor 2002; Frauenfelder in Int Math Res Notices 42:2179–2269, 2004; Bourgeois and Oancea in Duke Math J 146(1), 71–174, 2009). The homology of this complex is the symplectic homology of W (Diogo and Lisi in J Topol 12:966–1029, 2019). Let X be obtained from overline{W} by collapsing the boundary Y along Reeb orbits, giving a codimension two symplectic submanifold Sigma . Under monotonicity assumptions on X and Sigma , we show that for generic data, the differential in our chain complex counts elements of moduli spaces of cascades that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential.

Viterbo’s transfer morphism for symplectomorphisms

We construct an analogue of Viterbo’s transfer morphism for Floer homology of an automorphism of a Liouville domain. As an application we prove that the Dehn–Seidel twist along any Lagrangian sphere in a Liouville domain of dimension [Formula: see text] has infinite order in the symplectic mapping class group.

On partially wrapped Fukaya categories

We define a new class of symplectic objects called ‘stops’, which, roughly speaking, are Liouville hypersurfaces in the boundary of a Liouville domain. Locally, these can be viewed as pages of a compatible open book. To a Liouville domain with a collection of disjoint stops, we assign an A ∞ -category called its partially wrapped Fukaya category. An exact Landau–Ginzburg model gives rise to a stop, and the corresponding partially wrapped Fukaya category is meant to agree with the Fukaya category one is supposed to assign to the Landau–Ginzburg model. As evidence, we prove a formula that relates these partially wrapped Fukaya categories to the wrapped Fukaya category of the underlying Liouville domain. This operation is mirror to removing a divisor.

Volume growth in the component of fibered twists

For a Liouville domain [Formula: see text] whose boundary admits a periodic Reeb flow, we can consider the connected component [Formula: see text] of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, in the component [Formula: see text] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [Formula: see text] has infinite order in [Formula: see text] if there is an admissible Lagrangian [Formula: see text] in [Formula: see text] whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of [Formula: see text]-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse–Bott spectral sequences.

Maximum principles in symplectic homology

In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex structures than is usually considered. This allows us to extend the class of Hamiltonians which one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of infinite order in the symplectic mapping class group of a Liouville domain and prove existence results for translated points.

The positive equivariant symplectic homology as an invariant for some contact manifolds

We show that positive $S^1$-equivariant symplectic homology is a contact invariant for a subclass of contact manifolds which are boundaries of Liouville domains. In nice cases, when the set of Conley-Zehnder indices of all good periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the positive $S^1$-equivariant symplectic homology can be computed; it is generated by those orbits. We prove a Viterbo functoriality property: when one Liouville domain is embedded into an other one, there is a morphism (reversing arrows) between their positive $S^1$-equivariant symplectic homologies and morphisms compose nicely. These properties allow us to give a proof of Ustilovsky's result on the number of non isomorphic contact structures on the spheres $S^{4m+1}$. They also give a new proof of a Theorem by Ekeland and Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in $\mathbb{R}^{2n}$. We extend this result to some hypersurfaces in some negative line bundles.

Floer homology of automorphisms of Liouville domains

We introduce a combination of fixed point Floer homology and symplectic homology for Liouville domains. As an application, we detect non-trivial elements in the symplectic mapping class group of a Liouville domain.

Symmetric periodic orbits and uniruled real Liouville domains

A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.

Symplectic Tate homology

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.

The asymptotics of ECH capacities

In a previous paper, the second author used embedded contact homology (ECH) of contact three-manifolds to define “ECH capacities” of four-dimensional symplectic manifolds. In the present paper we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This follows from a more general theorem relating the volume of a contact three-manifold to the asymptotics of the amount of symplectic action needed to represent certain classes in ECH. The latter theorem was used by the first and second authors to show that every contact form on a closed three-manifold has at least two embedded Reeb orbits.

Legendrian realization in convex Lefschetz fibrations and convex stabilizations

Abstract We show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W, any Lagrangian submanifold with trivial first de Rham cohomology group, embedded on a (symplectic) page of the (induced) convex open book on ∂ W $\partial W$ , can be assumed to be Legendrian in ∂ W $\partial W$ with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. We also show that the convex stabilization of a compact convex Lefschetz fibration on W yields a compact convex Lefschetz fibration on a Liouville domain W' which is exact symplectomorphic to a positive expansion of W. In particular, with the induced structures ∂ W $\partial W$ and ∂ W ' $\partial W^{\prime }$ are contactomorphic.

Symplectic homology of displaceable Liouville domains and leafwise intersection points

In this note we prove that the symplectic homology of a Liouville domain W displaceable in the symplectic completion vanishes. Nevertheless if the Euler characteristic of (W,\p W) is odd, the filtered symplectic homologies of W do not vanish and give rise to leafwise intersection points on the symplectic completion of W for a perturbation displacing $W$ from itself. In contrast to the existing results we can find a leafwise intersection point for a given period but its energy varies by period instead.

Topological quantum field theory structure on symplectic cohomology

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic cohomology. These constructions yield new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. We prove that if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary of M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian lying in the boundary. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology. Finally, we use the TQFT to prove that symplectic cohomology vanishes iff Rabinowitz Floer cohomology vanishes.

Quantitative Embedded Contact Homology

Define a “Liouville domain” to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset thereof) a sequence of real numbers, which we call “ECH capacities”. The ECH capacities of a Liouville domain are defined in terms of the “ECH spectrum” of its boundary, which measures the amount of symplectic action needed to represent certain classes in embedded contact homology. Using cobordism maps on embedded contact homology (defined in joint work with Taubes), we show that the ECH capacities are monotone with respect to symplectic embeddings. We compute the ECH capacities of ellipsoids, polydisks, certain subsets of the cotangent bundle of $T^ 2$, and disjoint unions of examples for which the ECH capacities are known. The resulting symplectic embedding obstructions are sharp in some interesting cases, for example for the problem of embedding an ellipsoid into a ball (as shown by McDuff-Schlenk) or embedding a disjoint union of balls into a ball. We also state and present evidence for a conjecture under which the asymptotics of the ECH capacities of a Liouville domain recover its symplectic volume.