Lagrangian Submanifolds Research Articles

An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature

In this paper, we establish an optimal inequality involving normalized δ-Casorati curvature δC(n−1) of Lagrangian submanifolds in n-dimensional complex space forms. We derive a very singular and unexpected result: the lower bounds of the normalized δ-Casorati curvatures δC(n−1) and δCˆ(n−1) in terms of dimension, the holomorphic sectional curvature, the normalized scalar curvature and the squared mean curvature of the submanifold, are different, in contrast to all previous results obtained for several classes of submanifolds in many ambient spaces. We also investigate the equality case of the inequality and prove that a Casorati δC(n−1)-ideal Lagrangian submanifold of a complex space form without totally geodesic points is an H-umbilical Lagrangian submanifold of ratio 4. Some examples are discussed in the last part of the paper, showing that the constants in the inequality obtained in this work are the best possible.

Flexible Lagrangians

AbstractWe introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed n-manifolds of dimension n > 2 can be realized as exact Lagrangian submanifolds of $T^{\ast }S^n$ with possibly exotic Weinstein symplectic structures. These Weinstein structures on $T^{\ast } S^n$, infinitely many of which are distinct, are formed by a single handle attachment to the standard 2n-ball along the Legendrian boundaries of flexible Lagrangians. We also formulate a number of open problems.

Bifurcation of solutions to Hamiltonian boundary value problems

A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.

On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature – Submanifolds versions

In this paper, from the viewpoint of submanifold theory, we study the isolation phenomena of Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor. Firstly, for any locally strongly convex affine hyperspheres in an (n+1)-dimensional equiaffine space Rn+1 with constant scalar curvature, we prove an inequality involving the traceless Ricci tensor, the Pick invariant and the scalar curvature. The inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality. Secondly, and analogously, for Lagrangian minimal submanifolds of the complex projective space CPn equipped with the Fubini–Study metric, under the condition that the Weyl conformal curvature tensor vanishes, we establish a similar but reverse inequality involving the traceless Ricci tensor, the scalar curvature and the squared norm of the second fundamental form. The inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality.

The inverse problem of the calculus of variations for discrete systems

We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design and computer simulation of numerical integrators for constrained systems. For instance, nonholonomic mechanics is generally non variational but some special cases admit an alternative variational description. We apply some standard nonholonomic integrators to such an example to study which ones conserve this property.

Geometry of quantum dynamics in infinite-dimensional Hilbert space

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional, i.e. we do not restrict considerations to finite-dimensional Hilbert spaces, contrary to many other works on the geometry of quantum mechanics, and include a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we also obtain results concerning coadjoint orbits of the unitary group in infinite dimensions, embedding of pure states in the unitary group, and self-adjoint extensions of symmetric relations.

Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids

The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems, and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew’s triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the geometric formalism for mechanics on Lie algebroids developed by M. de Leon, J.C. Marrero and E. Martinez on “Lagrangian submanifolds and dynamics on Lie algebroids”.

Special Lagrangian and deformed Hermitian Yang–Mills on tropical manifold

From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mari\~no, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibration.

Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a K\"ahler-Einstein manifold to more general K\"ahler manifolds including a Fano manifold equipped with a K\"ahler form $\omega\in 2\pi c_1(M)$ by using the methodology proposed by T. Behrndt. Namely, we first consider a weighted measure on a Lagrangian submanifold $L$ in a K\"ahler manifold $M$ and investigate the variational problem of $L$ for the weighted volume functional. We call a stationary point of the weighted volume functional $f$-minimal, and define the notion of Hamiltonian $f$-stability as a local minimizer under Hamiltonian deformations. We show such examples naturally appear in a toric Fano manifold. Moreover, we consider the generalized Lagrangian mean curvature flow in a Fano manifold which is introduced by Behrndt and Smoczyk-Wang. We generalize the result of H. Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian deformation of an $f$-minimal and Hamiltonian $f$-stable Lagrangian submanifold, then the generalized MCF converges exponentially fast to an $f$-minimal Lagrangian submanifold.

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.

Rabinowitz Floer homology and mirror symmetry

We define a quantitative invariant of Liouville cobordisms with monotone filling through an action-completed symplectic cohomology theory. We illustrate the non-trivial nature of this invariant by computing it for annulus subbundles of the tautological bundle over C P 1 and give further conjectural computations based on mirror symmetry. We prove a non-vanishing result in the presence of Lagrangian submanifolds with non-vanishing Floer homology.

A Hamilton–Jacobi theory for implicit differential systems

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of TT*Q generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton–Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures.

Lagrangian submanifolds with constant angle functions of the nearly Kähler S3×S3

We study Lagrangian submanifolds of the nearly Kähler S 3 × S 3 with respect to their so called angle functions. We show that if all angle functions are constant, then the submanifold is either totally geodesic or has constant sectional curvature and there is a classification theorem that follows from Dioos et al. (2018). Moreover, we show that if precisely one angle function is constant, then it must be equal to 0 , π 3 or 2 π 3 . Using then two remarkable constructions together with the classification of Lagrangian submanifolds of which the first component has nowhere maximal rank from, Bektaş et al. (2018), we obtain a classification of such Lagrangian submanifolds.

Kähler structures on spaces of framed curves

We consider the space $${\mathcal {M}}$$ of Euclidean similarity classes of framed loops in $${\mathbb {R}}^3$$ . Framed loop space is shown to be an infinite-dimensional Kahler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of $${\mathcal {M}}$$ by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of $$\mathcal {M}$$ explicitly. Using this description, we show that $${\mathcal {M}}$$ and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in $${\mathcal {M}}$$ as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on $${\mathcal {M}}$$ is shown to be Hamiltonian, and this is used to characterize the critical points of the weighted total twist functional.

Integration over families of Lagrangian submanifolds in BV formalism

Gauge fixing is interpreted in BV formalism as a choice of Lagrangian submanifold in an odd symplectic manifold (the BV phase space). A natural construction defines an integration procedure on families of Lagrangian submanifolds. In string perturbation theory, the moduli space integrals of higher genus amplitudes can be interpreted in this way. We discuss the role of gauge symmetries in this construction. We derive the conditions which should be imposed on gauge symmetries for the consistency of our integration procedure. We explain how these conditions behave under the deformations of the worldsheet theory. In particular, we show that integrated vertex operator is actually an inhomogeneous differential form on the space of Lagrangian submanifolds.

Pseudo-symmetries of generalized Wintgen ideal Lagrangian submanifolds

Mihai obtained the Wintgen inequality, also known as the generalized Wintgen inequality, for Lagrangian submanifolds in complex space forms and also characterized the corresponding equality case. Submanifolds M which satisfy the equality in these optimal general inequalities are called generalized Wintgen ideal submanifolds in the ambient space ?M. For generalized Wintgen ideal Lagrangian submanifolds Mn in complex space forms ?Mn(4c), we will show some properties concerning different kinds of their pseudosymmetry in the sense of Deszcz.

A note on the normalization of generating functions

In the present note, I will propose some insights on the normalization of generating functions for Lagrangian submanifolds. From the literature (see, for example [4], [6], [7], [3] and [1]), it is clear that a problem exists concerning the nonuniqueness of generating functions and, in particular, of the generating functions quadratic at infinity (GFQI). This problem can be avoided introducing a normalization on the whole set of generating functions that will allow us to(ⅰ) choose an unique GFQI for Lagrangian submanifolds of the form $\varphi(L)$, where $L$ is a Lagrangian submanifold and $\varphi$ is an Hamiltonian isotopy;(ⅱ) compare the critical values $c(α, S_1)$ and $c(α, S_2)$ of two GFQI generating the submanifolds, $\varphi_1(L)$ and $\varphi_2(L)$, where $\varphi_1$ and $\varphi_2$ are Hamiltonian isotopies relative to two Hamiltonians $H_1$ and $H_2$, respectively.

On totally real statistical submanifolds

In the present paper, first we prove some results by using fundamental properties of totally real statistical submanifolds immersed into holomorphic statistical manifolds. Further, we obtain the generalizedWintgen inequality for Lagrangian statistical submanifolds of holomorphic statistical manifolds with constant holomorphic sectional curvature c. The paper finishes with some geometric consequences of obtained results.

Cubic planar graphs and Legendrian surface theory

We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings. We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte-Gabella-Goncharov [DGGo]. We exploit this structure to find conjectural open Gromov-Witten invariants for the non-exact filling, following Aganagic-Vafa [AV, AV2].

Morita equivalence and the generalized Kähler potential

We solve the problem of determining the fundamental degrees of freedom underlying a generalized Kähler structure of symplectic type. For a usual Kähler structure, it is well-known that the geometry is determined by a complex structure, a Kähler class, and the choice of a positive $(1, 1)$-form in this class, which depends locally on only a single real-valued function: the Kähler potential. Such a description for generalized Kähler geometry has been sought since it was discovered in 1984. We show that a generalized Kähler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on only a single real-valued function, which we call the generalized Kähler potential. Our solution draws upon, and specializes to, the many results in the physics literature which solve the problem under the assumption (which we do not make) that the Poisson structures involved have constant rank. To solve the problem we make use of, and generalize, two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds; the second is Donaldson’s interpretation of a Kähler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.