Hamiltonian Isotopy Research Articles

Lagrangian Surplusection Phenomena

Suppose you have a family of Lagrangian submanifolds $L_t$ and an auxiliary Lagrangian $K$. Suppose that $K$ intersects some of the $L_t$ more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of $K$? Or will any Lagrangian isotopic to $K$ surplusect some of the fibres? We argue that in several important situations, surplusection cannot be eliminated, and that a better understanding of surplusection phenomena (better bounds and a clearer understanding of how the surplusection is distributed in the family) would help to tackle some outstanding problems in different areas, including Oh's conjecture on the volume-minimising property of the Clifford torus and the concurrent normals conjecture in convex geometry. We pose many open questions.

Hofer distance on Lagrangian links inside the disc

We show that the set of Hamiltonian isotopies of certain unions of circles inside the disc is unbounded for the Hofer distance. The proof relies on a result by Francesco Morabito (Link Floer Homology and a Hofer Pseudometric on Braids. https://arxiv.org/abs/2301.08105, 2024) together with a standard argument of Michael Khanevsky (Geometric and topological aspects of Lagrangian submanifolds-intersections, diameter and Floer theory. PhD thesis. Tel-Aviv University, 2011).

Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving

Abstract Given a closed, orientable Lagrangian submanifold L in a symplectic manifold (X, ω), we show that if L is relatively exact then any Hamiltonian diffeomorphism preserving L setwise must preserve its orientation. In contrast to previous results in this direction, there are no spin hypotheses on L. Curiously, the proof uses only mod 2 coefficients in its singular and Floer cohomology rings.

Geometry of symplectic flux and Lagrangian torus fibrations

AbstractSymplectic flux measures the areas of cylinders swept in the process of a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangian submanifold that we define using its Fukaya algebra. The main geometric feature of the invariant is its concavity over isotopies with linear flux. We derive constraints on flux, Weinstein neighbourhood embeddings and holomorphic disk potentials for Gelfand–Cetlin fibres of Fano varieties in terms of their polytopes. We also describe the space of fibres of almost toric fibrations on the complex projective plane up to Hamiltonian isotopy, and provide other applications.

Exact Lagrangians in four‐dimensional symplectisations

AbstractIn this note, we provide explicit constructions of exact Lagrangian embeddings of tori and Klein bottles inside the symplectisation of an overtwisted contact three‐manifold. Note that any closed exact Lagrangian in the symplectisation is displaceable by a Hamiltonian isotopy. We also use positive loops to exhibit elementary examples of topologically linked Legendrians that are dynamically non‐interlinked in the sense of Entov–Polterovich.

Families of relatively exact Lagrangians, free loop spaces and generalised homology

AbstractWe prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy $$\psi ^1$$ ψ 1 of a symplectic manifold $$(M, \omega )$$ ( M , ω ) fixing a relatively exact Lagrangian L setwise must act trivially on $$R_*(L)$$ R ∗ ( L ) , where $$R_*$$ R ∗ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over $${\mathbb {Z}}/2$$ Z / 2 and over $${\mathbb {Z}}$$ Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that $$\psi ^1|_L$$ ψ 1 | L acts trivially on $$R_*({\mathcal {L}}L)$$ R ∗ ( L L ) , where $${\mathcal {L}}L$$ L L is the free loop space of L. From this we deduce that when L is a surface or a $$K(\pi , 1)$$ K ( π , 1 ) , $$\psi ^1|_L$$ ψ 1 | L is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over $${\mathbb {Z}}/2$$ Z / 2 .

Lagrangian cobordisms and Lagrangian surgery

Lagrangian k -surgery modifies an immersed Lagrangian submanifold by topological k -surgery while removing a self-intersection. Associated to a k -surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. This exact homotopy can be chosen with as small Hofer norm as desired. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated with the handle attachment to the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds. In an appendix, we describe a 2-ended embedded monotone Lagrangian cobordism which is not the suspension of a Hamiltonian isotopy following a suggestion of Abouzaid and Auroux.

Lagrangian fillings in $A$-type and their Kálmán loop orbits

We compare two constructions of exact Lagrangian fillings of Legendrian positive braid closures, the Legendrian weaves of Casals–Zaslow, and the decomposable Lagrangian fillings of Ekholm–Honda–Kálmán, and show that they coincide for large families of Lagrangian fillings. As a corollary, we obtain an explicit correspondence between Hamiltonian isotopy classes of decomposable Lagrangian fillings of Legendrian (2,n) torus links described by Ekholm–Honda–Kálmán and the weave fillings constructed by Treumann and Zaslow. We apply this result to describe the orbital structure of the Kálmán loop and give a combinatorial criteria to determine the orbit size of a filling. We follow our geometric discussion with a Floer-theoretic proof of the orbital structure, where an identity studied by Euler in the context of continued fractions makes a surprise appearance. We conclude by giving a purely combinatorial description of the Kálmán loop action on the fillings discussed above in terms of edge flips of triangulations.

New lower bounds on the number of intersections of monotone Lagrangian submanifolds

Considering a closed monotone Lagrangian submanifold L , L, we give, under some hypotheses, a lower bound on the intersection number of L L with its image by a generic Hamiltonian isotopy. First for monotone Lagrangian submanifolds L L which are K ( π , 1 ) \mathbf {K}(\pi ,1) and, in particular, for monotone Lagrangian submanifolds with negative sectional curvature this bound is 1+ β 1 ( L ) . \beta _{1}(L). In more general cases the lower bound is weaker. We generalise some results previously obtained by L. Buhovsky in [J. Topol. Anal. 2 (2010), pp. 57–75] and P. Biran and O. Cornea in [Geom. Topol. 13 (2009), pp. 2881–2989].

Tangles, relative character varieties, and holonomy perturbed traceless flat moduli spaces

We prove that the restriction map from the subspace of regular points of the holonomy perturbed SU(2) traceless flat moduli space of a tangle in a 3-manifold to the traceless flat moduli space of its boundary marked surface is a Lagrangian immersion. A key ingredient in our proof is the use of composition in the Weinstein category, combined with the fact that SU(2) holonomy perturbations in a cylinder induce Hamiltonian isotopies. In addition, we show that $(S^2,4)$, the 2-sphere with four marked points, is its own traceless flat SU(2) moduli space.

C0-transport of flux geometry

The goal of this paper is to study, in a large scale point of view, the flux geometry of a closed symplectic manifold (M,ω): namely, the topological counterpart of the flux homomorphism. Using metrics arising from the decomposition of closed 1-forms with respect to an arbitrary linear section S, we generalize the construction of the group of strong symplectic homeomorphisms. The flux homomorphism for symplectomorphisms is extended to a surjective group homomorphism Sω0 on the group of S-homeomorphisms. We prove that the kernel of Sω0 is path connected, coincides with the subgroup Hameo(M,ω) of all Hamiltonian homeomorphisms and investigate the discreteness of the corresponding flux group SΓω. Later on, without appealing to any lifting map, we give an alternative proof of a result from the classical flux geometry saying that any smooth symplectic isotopy in Ham(M,ω) is a Hamiltonian isotopy. Furthermore under some hypothesis, we prove that any S-topological isotopy in Hameo(M,ω) is a continuous Hamiltonian isotopy. We also proved that any S-topological isotopy with trivial flux is homotopic to a continuous Hamiltonian isotopy, relatively to fixed endpoints.

Braid loops with infinite monodromy on the Legendrian contact DGA

We present the first examples of elements in the fundamental group of the space of Legendrian links in ( S 3 , ξ st ) $(\mathbb {S}^3,\xi _{\text{st}})$ whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These new families include the first-known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, and in particular give the first Floer-theoretic proof that Legendrian ( n , m ) $(n,m)$ torus links have infinitely many Lagrangian fillings if n ⩾ 3 , m ⩾ 6 $n\geqslant 3,m\geqslant 6$ or ( n , m ) = ( 4 , 4 ) , ( 4 , 5 ) $(n,m)=(4,4),(4,5)$ . In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus in the same smooth isotopy class, but distinct Hamiltonian isotopy classes. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.

A note on the infinite number of exact Lagrangian fillings for spherical spuns

In this short note we discuss high-dimensional examples of Legendrian submanifolds of the standard contact Euclidean space with an infinite number of exact Lagrangian fillings up to Hamiltonian isotopy. They are obtained from the examples of Casals and Ng by applying to them the spherical spinning construction.

Families of Legendrians and Lagrangians with unbounded spectral norm

Viterbo has conjectured that any Lagrangian in the unit co-disc bundle of a torus which is Hamiltonian isotopic to the zero-section satisfies a uniform bound on its spectral norm; a recent result by Shelukhin showed that this is indeed the case. The modest goal of our note is to explore two natural generalisations of this geometric setting in which the bound of the spectral norm fails: first, passing to Legendrian isotopies in the contactisation of the unit co-disc bundle (recall that any Hamiltonian isotopy can be lifted to a Legendrian isotopy) and, second, considering Hamiltonian isotopies but after modifying the co-disc bundle by attaching a critical Weinstein handle.

Deformations of Lagrangian submanifolds in log-symplectic manifolds

This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that the gauge equivalence relation of the DGLA corresponds with the geometric notion of equivalence by Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove a rigidity statement for the more flexible equivalence relation by Poisson isotopies.

Caustics of Lagrangian homotopy spheres with stably trivial Gauss map

For each positive integer $n$, we give a geometric description of the stably trivial elements of the group $\pi_n U_n/O_n$. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.

Families of monotone Lagrangians in Brieskorn\u2013Pham hypersurfaces

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn–Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian S^1 times Sigma _g in {mathbb {C}}^3, distinguished by soft invariants for any genus g ge 2; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn–Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.

Linking of Lagrangian Tori and Embedding Obstructions in Symplectic 4-Manifolds

Abstract We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a 1st corollary, we strengthen a result due independently to Eliashberg–Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$, which are homologous to the zero section. As a 2nd corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the 2nd part of the paper, we study linking of Lagrangian tori in $({\mathbb{R}}^4, \omega )$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.

Note on coisotropic Floer homology and leafwise fixed points

<p style='text-indent:20px;'>For an adiscal or monotone regular coisotropic submanifold <inline-formula><tex-math id="M1">$ N $</tex-math></inline-formula> of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of <inline-formula><tex-math id="M2">$ N $</tex-math></inline-formula>. Given a Hamiltonian isotopy <inline-formula><tex-math id="M3">$ \varphi = ( \varphi^t) $</tex-math></inline-formula> and a suitable almost complex structure, the corresponding Floer chain complex is generated by the <inline-formula><tex-math id="M4">$ (N, \varphi) $</tex-math></inline-formula>-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.</p><p style='text-indent:20px;'>Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.</p><p style='text-indent:20px;'>The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.</p>

On a local systolic inequality for odd-symplectic forms

The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let \Omega be an odd-symplectic form on an oriented closed manifold \Sigma of odd dimension. We say that \Omega is Zoll if the trajectories of the flow given by \Omega are the orbits of a free S^1 -action. After defining the volume of \Omega and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided \Omega is Zoll. This builds the equality case of a conjectural systolic inequality for odd-symplectic forms close to a Zoll one. We prove the conjecture when the S^1 -action yields a flat S^1 -bundle or when \Omega is quasi-autonomous. Together with previous work [BK19a], this establishes the conjecture in dimension three. This new inequality recovers the local contact systolic inequality (recently proved in [AB19]) as well as the inequality between the minimal action and the Calabi invariant for Hamiltonian isotopies C^1 -close to the identity on a closed symplectic manifold. Applications to the study of periodic magnetic geodesics on closed orientable surfaces is given in the companion paper [BK19b].