Lagrangian Submanifolds Research Articles

Stable generalized complex structures

A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi–Yau structure. We introduce a formalism which allows us to view such structures as symplectic forms with singularities of logarithmic or elliptic type. This allows us to define two period maps: one for deformations in which the background 3-form flux is fixed, and one for which the flux is allowed to vary. As a result, we prove the unobstructedness of each of these deformation problems. We use the same approach to establish local classification theorems for the degeneracy locus as well as for analogues of Lagrangian submanifolds called Lagrangian branes.

Reductions of minimal Lagrangian submanifolds with symmetries

Let M be a Fano manifold equipped with a Kahler form $$\omega \in 2\pi c_1(M)$$ and K a connected compact Lie group acting on M as holomorphic isometries. In this paper, we show the minimality of a K-invariant Lagrangian submanifold L in M with respect to a globally conformal Kahler metric is equivalent to the minimality of the reduced Lagrangian submanifold $$L_0=L/K$$ in a Kahler quotient $$M_0$$ with respect to the Hsiang–Lawson metric. Furthermore, we give some examples of Kahler reductions by using a circle action obtained from a cohomogenenity one action on a Kahler–Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.

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.

Classification of transversal Lagrangian stars

A Lagrangian star is a system of three Lagrangian submanifolds of the symplectic space intersecting at a common point. In this work we classify transversal Lagrangian stars in the symplectic space in the analytic category under the action of symplectomorphisms by using the method of algebraic restrictions. We present a list of all transversal Lagrangian star.

Classification of δ(2,n − 2)-ideal Lagrangian submanifolds in n-dimensional complex space forms

It was proven in [4] that every Lagrangian submanifold M of a complex space form M˜n(4c) of constant holomorphic sectional curvature 4c satisfies the following optimal inequality:δ(2,n−2)≤n2(n−2)4(n−1)H2+2(n−2)c, where H2 is the squared mean curvature and δ(2,n−2) is a δ-invariant on M. In this paper we classify Lagrangian submanifolds of complex space forms M˜n(4c), n≥5, which satisfy the equality case of this improved inequality at every point.

Open Gromov-Witten Invariants from the Augmentation Polynomial

A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = O P 1 ( − 1 , − 1 ) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds L K ⊂ X obtained from the conormal bundles of knots K ⊂ S 3 . This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For ( r , s ) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the ( 3 , 2 ) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants.

Fourier integral operators on Lie groupoids

As announced in [36], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study Lagrangian conic submanifolds of the symplectic groupoid T⁎G. This includes their product, transposition and inversion. We also study the relationship between these Lagrangian submanifolds and the equivariant families of Lagrangian submanifolds of T⁎Gx×T⁎Gx parametrized by the units x∈G(0) of G. This allows us to select a subclass of Lagrangian distributions on any Lie groupoid G that deserve the name of Fourier integral G-operators (G-FIOs). By construction, the class of G-FIOs contains the class of equivariant families of ordinary Fourier integral operators on the manifolds Gx, x∈G(0). We then develop for G-FIOs the first stages of the calculus in the spirit of Hormander's work. Finally, we illustrate this calculus in the case of manifolds with boundary.

Seidel elements and potential functions of holomorphic disc counting

Let $M$ be a symplectic manifold equipped with a Hamiltonian circle action and let $L$ be an invariant Lagrangian submanifold of $M$. We study the problem of counting holomorphic disc sections of the trivial $M$-bundle over a disc with boundary in $L$ through degeneration. We obtain a conjectural relationship between the potential function of $L$ and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng [8, 9] relating certain correction terms appearing in the Seidel elements with the potential function.

Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry–Mather theory

AbstractWe construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants, developed by Oh, and results, by Abouzaid, about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalisation of graph selector. We then, following Bernard–Oliveira dos Santos, use these results to give a new characterisation of the Aubry and Mañé sets of a Tonelli Hamiltonian and to generalise a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

Quantum Reidemeister Torsion, Open Gromov–Witten Invariants and a Spectral Sequence of OH

We adapt classical Reidemeister torsion to monotone Lagrangian submanifolds using the pearl complex of Biran and Cornea. The definition involves generic choices of data and we identify a class of Lagrangians for which this torsion is invariant and can be computed in terms of genus zero open Gromov-Witten invariants. This class is defined by a vanishing property of a spectral sequence of Oh in Lagrangian Floer theory.

Second variation for L-minimal Legendrian submanifolds in pseudo-Sasakian manifolds

In this paper we provide a second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian-Sasakian manifolds and relate the L-stability of L-minimal Legendrians in a Sasakian M to their L-stability in an associated Lorentzian-Sasakian structure on M.

The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the L-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the L-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from k initial scalar L-transforms of a given submanifold of constant curvature, a whole k-dimensional cube all of whose remaining $$2^k-(k+1)$$ vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchi-cube theorems, for the classes of n-dimensional flat Lagrangian submanifolds of $${\mathbb {C}}^n$$ and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space $${\mathbb {C}}{\mathbb {P}}^n(4c)$$ or the complex hyperbolic space $${\mathbb {C}}{\mathbb {H}}^n(4c)$$ of complex dimension n and constant holomorphic curvature 4c.

Families of gauge conditions in BV formalism

In BV formalism we can consider a Lagrangian submanifold as a gauge condition. Starting with the BV action functional we construct a closed form on the space of Lagrangian submanifolds. If the action functional is invariant with respect to some group H and Λ is an H-invariant family of Lagrangian submanifold then under certain conditions we construct a form on Λ that descends to a closed form on Λ/H. Integrating the latter form over a cycle in Λ/H we obtain numbers that can have interesting physical meaning. We show that one can get string amplitudes this way. Applying this construction to topological quantum field theories one obtains topological invariants.

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$ S 3 × S 3

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kahler $$\mathbb {S}^3 \times \mathbb {S}^3$$ can only be $$\frac{2}{\sqrt{3}}$$ or $$\frac{4}{\sqrt{3}}$$ .

A new structural approach to isoparametric hypersurfaces in spheres

In this paper we show that the long-standing problem of classifying all isoparametric hypersurfaces in spheres with six different principal curvatures is still not complete. Moreover, we develop a structural approach that may be helpful for such a classification. Instead of working with the isoparametric hypersurface family in the sphere, we consider the associated Lagrangian submanifold of the real Grassmannian of oriented 2-planes in {mathbb {R}}^{n+2}. We obtain new geometric insights into classical invariants and identities in terms of the geometry of the Lagrangian submanifold.

Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations

This survey presents a generalization of the notion of a toric structure on a compact symplectic manifold: the notion of a pseudotoric structure. The language of these new structures appears to be a convenient and natural tool for describing many non-standard Lagrangian submanifolds and cycles (Chekanov's exotic tori, Mironov's cycles in certain particular cases, and others) as well as for constructing Lagrangian fibrations (for example, special fibrations in the sense of Auroux on Fano varieties). Known properties of pseudotoric structures and constructions based on these properties are discussed, as well as open problems whose solution may be of importance in symplectic geometry and mathematical physics. Bibliography: 28 titles.

Admissible boundary conditions for Hamiltonian field theories

In the setting of a multisymplectic formalism for Hamiltonian theories on manifolds with boundary a class of admissible boundary conditions based on the principle of preservation of the gauge symmetries of the theory is presented. Such admissible conditions are characterized as those boundary conditions determined by Lagrangian submanifolds on the space of fields at the boundary lying in the zero level set of the moment map of the group of gauge transformations at the boundary. The examples of gauge covariant fields and pure Yang–Mills theories are analyzed.

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.

The Künneth theorem for the Fukaya algebra of a product of Lagrangians

Given a compact Lagrangian submanifold [Formula: see text] of a symplectic manifold [Formula: see text], Fukaya, Oh, Ohta and Ono construct a filtered [Formula: see text]-algebra [Formula: see text], on the cohomology of [Formula: see text], which we call the Fukaya algebra of [Formula: see text]. In this paper, we describe the Fukaya algebra of a product of two Lagrangians submanifolds [Formula: see text]. Namely, we show that [Formula: see text] is quasi-isomorphic to [Formula: see text], where [Formula: see text] is the tensor product of filtered [Formula: see text]-algebras defined in [L. Amorim, Tensor product of filtered [Formula: see text]-algebras, J. Pure Appl. Algebra 220(12) (2016) 3984–4016]. As a corollary of this quasi-isomorphism, we obtain a description of the bounding cochains on [Formula: see text] and of the Floer cohomology of [Formula: see text].

On the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms

The generalized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for submanifolds in real space forms. It is also known as the DDVV conjecture. It was proven recently by Lu (2011) and by Ge and Tang (2008), independently. The present author established a generalized Wintgen inequality for Lagrangian submanifolds in complex space forms in 2014. In the present paper we obtain the DDVV inequality, also known as generalized Wintgen inequality, for Legendrian submanifolds in Sasakian space forms. Some geometric applications are derived. Also we state such an inequality for contact slant submanifolds in Sasakian space forms.