Lagrangian Embeddings Research Articles

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.

Monotone Lagrangian submanifolds of ℂn and toric topology

Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside $\mathbb{C}^n$. They associated a closed embedded Lagrangian $L$ to each Delzant polytope $P$. In this paper we develop their ideas and prove that $L$ is monotone if and only if the polytope $P$ is Fano. In some examples, we further compute the minimal Maslov numbers. Namely, let $\mathcal{N}\to T^k$ be some fibration over the $k$-dimensional torus with a fiber equal to either $S^k \times S^l$, or $S^k \times S^l \times S^m$, or $\#_5(S^{2p-1} \times S^{n-2p-2})$. We construct monotone Lagrangian embeddings $\mathcal{N} \subset \mathbb{C}^n$ with different minimal Maslov number, and therefore distinct up to Lagrangian isotopy. Moreover, we show that some of our embeddings are smoothly isotopic but not Lagrangian isotopic.

Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces

Abstract A Kähler-type form is a symplectic form compatible with an integrable complex structure. Let $M$ be either a torus or a K3-surface equipped with a Kähler-type form. We show that the homology class of any Maslov-zero Lagrangian torus in $M$ has to be nonzero and primitive. This extends previous results of Abouzaid and Smith (for tori) and Sheridan and Smith (for K3-surfaces) who proved it for particular Kähler-type forms on $M$. In the K3 case, our proof uses dynamical properties of the action of the diffeomorphism group of $M$ on the space of the Kähler-type forms. These properties are obtained using Shah’s arithmetic version of Ratner’s orbit closure theorem.

The rectangular peg problem

For every smooth Jordan curve $\gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $\gamma$. The proof relies on the theorem of Shevchishin and Nemirovski that the Klein bottle does not admit a smooth Lagrangian embedding in $\mathbb{C}^2$.

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.

A log PSS morphism with applications to Lagrangian embeddings

Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of distinguished classes in symplectic cohomology of the complement $X = M \backslash \mathbf{D}$. Under further "topological" assumptions on the pair, these classes can be organized into a Log(arithmic) PSS morphism, from a vector space which we term the logarithmic cohomology of $(M, \mathbf{D})$ to symplectic cohomology. Turning to applications, we show that these methods and some knowledge of Gromov-Witten invariants can be used to produce dilations and quasi-dilations (in the sense of Seidel-Solomon [SS]) in examples such as conic bundles. In turn, the existence of such elements imposes strong restrictions on exact Lagrangian embeddings, especially in dimension 3. For instance, we prove that any exact Lagrangian in a complex 3-dimensional conic bundle over $(\mathbb{C}^*)^2$ must be diffeomorphic to $T^3$ or a connect sum $\#^n S^1 \times S^2$.

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>

Disjoinable Lagrangian tori and semisimple symplectic cohomology

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-downs and flips. As a consequence, there are many non-toric (non-compact) monotone symplectic manifolds whose wrapped Fukaya categories are proper.

Euler-like vector fields, normal forms, and isotropic embeddings

As shown by Bursztyn et al. (2019), germs of tubular neighborhood embeddings for submanifolds N⊆M are in one–one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of ‘normal forms results’ for geometric structures to the construction of an Euler-like vector field compatible with the given structure. We illustrate this principle in a variety of examples, including the Morse–Bott lemma, Weinstein’s Lagrangian embedding theorem, and Zung’s linearization theorem for proper Lie groupoids. In the second part of this article, we extend the theory to a weighted context, with an application to isotropic embeddings.

An application of the h-principle to manifold calculus

Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor $$\mathrm {Emb}_{\mathrm {Lag}}(-,N)$$ is the totally real embeddings functor $$\mathrm {Emb}_{\mathrm {TR}}(-,N)$$. More generally, for subsets $${\mathcal {A}}$$ of the m-plane Grassmannian bundle $${{\,\mathrm{{Gr}}\,}}(m,TN)$$ for which the h-principle holds for $${\mathcal {A}}$$-directed embeddings, we prove the analyticity of the $${\mathcal {A}}$$-directed embeddings functor $${{\,\mathrm{Emb}\,}}_{{\mathcal {A}}}(-,N)$$.

Symplectic triangle inequality

We prove a non-squeezing result for Lagrangian embeddings of the real projective plane into blow-ups of the symplectic ball.

On Lagrangian embeddings of closed nonorientable 3–manifolds

We prove that for any compact orientable connected 3-manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open 2-disk removed admits a Lagrangian embedding into the standard symplectic 6-space. Moreover, minimal Maslov number of the Lagrangian embedding is equal to 1.

C0-characterization of symplectic and contact embeddings and Lagrangian rigidity

We present a novel [Formula: see text]-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of [Formula: see text]-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from [Formula: see text]-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a [Formula: see text]-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of [Formula: see text]-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

Nearby Lagrangian fibers and Whitney sphere links

Let $n>3$, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ outside a compact set. Assume that $L$ is disjoint from the cotangent fiber at the origin. The projection of $L$ to the base extends to a map of the $n$-sphere $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.

Markov numbers and Lagrangian cell complexes in the complex projective plane

We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\ldots,N$, cannot be made disjoint unless $N\leq 3$ and the $p_i$ form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a $\mathbf{Q}$-Gorenstein smoothing whose general fibre is $\mathbf{CP}^2$.

Punctured holomorphic curves and Lagrangian embeddings

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.

Lagrangian exotic spheres

Let [Formula: see text]. We prove that the cotangent bundles [Formula: see text] and [Formula: see text] of oriented homotopy [Formula: see text]-spheres [Formula: see text] and [Formula: see text] are symplectomorphic only if [Formula: see text], where [Formula: see text] denotes the group of oriented homotopy [Formula: see text]-spheres under connected sum, [Formula: see text] denotes the subgroup of those that bound a parallelizable [Formula: see text]-manifold, and where [Formula: see text] denotes [Formula: see text] with orientation reversed. We further show that if [Formula: see text] and [Formula: see text] admits a Lagrangian embedding in [Formula: see text], then [Formula: see text]. The proofs build on [1] and [18] in combination with a new cut-and-paste argument; that also yields some interesting explicit exact Lagrangian embeddings, for instance of the sphere [Formula: see text] into the plumbing [Formula: see text] of cotangent bundles of certain exotic spheres. As another application, we show that there are re-parametrizations of the zero-section in the cotangent bundle of a sphere that are not Hamiltonian isotopic (as maps rather than as submanifolds) to the original zero-section.

On Lagrangian embeddings into the complex projective spaces

We prove that for any closed orientable connected [Formula: see text]-manifold [Formula: see text] and any Lagrangian immersion of the connected sum [Formula: see text] either into the complex projective [Formula: see text]-space [Formula: see text] or into the product [Formula: see text] of the complex projective line and the complex projective plane, there exists a Lagrangian embedding which is homotopic to the initial Lagrangian immersion. To prove this, we show that Eliashberg–Murphy’s [Formula: see text]-principle for Lagrangian embeddings with a concave Legendrian boundary and Ekholm–Eliashberg–Murphy–Smith’s [Formula: see text]-principle for self-transverse Lagrangian immersions with the minimal or near-minimal number of double points hold for six-dimensional simply connected compact symplectic manifolds.

On the immersion classes of nearby Lagrangians: Table 1.

We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.

On the Strict Arnold Chord Property and Coisotropic Submanifolds of Complex Projective Space

Let be a contact form on a manifoldM, and L M a closed Legendrian submanifold. I prove that L intersects some characteristic for at least twice if all characteristics are closed and of the same period, and embeds nicely into the product of R2n and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most /2. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.