Maslov Class Research Articles

Notes on a Category-Theoretic Definition of Maslov Classes of a Lagrangian Manifold

Notes on a Category-Theoretic Definition of Maslov Classes of a Lagrangian Manifold

Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $$W=\mathbb{T}^*(M)$$ of the Cotangent Bundle”

Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $$W=\mathbb{T}^*(M)$$ of the Cotangent Bundle”

On the Gauss map of equivariant immersions in hyperbolic space

Given an oriented immersed hypersurface in hyperbolic space H n + 1 $\mathbb {H}^{n+1}$ , its Gauss map is defined with values in the space of oriented geodesics of H n + 1 $\mathbb {H}^{n+1}$ , which is endowed with a natural para-Kähler structure. In this paper, we address the question of whether an immersion G $G$ of the universal cover of an n $n$ -manifold M $M$ , equivariant for some group representation of π 1 ( M ) $\pi _1(M)$ in Isom ( H n + 1 ) $\mathrm{Isom}(\mathbb {H}^{n+1})$ , is the Gauss map of an equivariant immersion in H n + 1 $\mathbb {H}^{n+1}$ . We fully answer this question for immersions with principal curvatures in ( − 1 , 1 ) $(-1,1)$ : while the only local obstructions are the conditions that G $G$ is Lagrangian and Riemannian, the global obstruction is more subtle, and we provide two characterizations, the first in terms of the Maslov class, and the second (for M $M$ compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.

-rigidity of Lagrangian submanifolds and punctured holomorphic disks in the cotangent bundle

AbstractOur main result is the $\mathbb {\mathcal {C}}^{0}$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

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.

Products and connected sums of spheres as monotone Lagrangian submanifolds

We obtain new restrictions on Maslov classes of monotone Lagrangian submanifolds of ℂn. We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp in certain cases.

Rational equivalence and Lagrangian tori on K3 surfaces

Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L \subset X to the skyscraper sheaf of a point y \in Y . We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the “Beauville–Voisin subring” in the Chow groups of Y , and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.

Knots in Mathematics and Medicine

The analogy between knots in mathematical knot theory and “knots” in medicine is studied for the example of a certain invariant of Legendrian knots. The connection between the “Chern class” and the “Maslov class” is analyzed. A conjecture on the connection between the “Connes-Chern character” and the “Maslov class” is proposed.

Involutions, obstructions and mirror symmetry

Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya A∞ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Floer cohomology approaches to mirror symmetry.More generally, our result continues to hold if the Lagrangian has cohomology the free graded algebra on a graded vector space V concentrated in odd degree, and the anti-symplectic involution acts on the cohomology of the Lagrangian by the induced map of negative the identity on V. It suffices for the Maslov class to vanish modulo 4.

On the Lagrangian Angle and the Kähler Angle of Immersed Surfaces in the Complex Plane ℂ2

In this paper, we discuss the Lagrangian angle and the Kahler angle of immersed surfaces in ℂ2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in ℂ2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kahler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kahler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kahler angle which imply rigidity theorems of self-shrinkers with Kahler angle under the condition that $${\smallint _M}{\left| h \right|^2}{{\rm{e}}^{ - {{{{\left| x \right|}^2}} \over 2}}}{\rm{d}}{V_M}\; < \;\infty $$ , where h and x denote, respectively, the second fundamental form and the position vector of the surface.

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.

Generalized Lagrangian mean curvature flows: The cotangent bundle case

Abstract In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel ( n , 0 ) {(n,0)} -form, just like the Calabi–Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also obtained by Nadler). The ideas in that paper were extended by Abouzaid who proved that vanishing Maslov class alone implies homotopy equivalence. In this paper we present a short proof of the fact that any exact Lagrangian with vanishing Maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism. When the fundamental group of the base is pro-finite this implies homotopy equivalence.

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

The theory of self-adjoint extensions of first- and second-order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space [Formula: see text] of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's theorem characterizing the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions [Formula: see text] is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of [Formula: see text] is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results. A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved. The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grassmannian is obtained. Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of self-adjoint extensions.

Constructing exact Lagrangian immersions with few double points

We establish, as an application of the results from Eliashberg and Murphy (Lagrangian caps, 2013), an h-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space \({\mathbb{R}^6_{\rm st}}\) with exactly one transverse double point. Our construction also yields a Lagrangian embedding \({S^1 \times S^2 \to \mathbb{R}^6_{\rm st}}\) with vanishing Maslov class.

Hamiltonian mean curvature flow

Let (Σ ,ω ) be a compact Riemann surface with constant curvature c. In this work, we proved that the mean curvature flow of a given Hamiltonian diffeomorphism on Σ provides a smooth path in Ham(Σ), the group of all Hamiltonian diffeomorphisms of Σ. This result gives a proof, in the case of graph of Hamiltonian diffeomorphisms to the conjecture of Thomas and Yau asserting that the mean curvature flow of a compact embedded Lagrangian submanifold S with zero Maslov class in a Calabi-Yau manifolds M exists for all time and converges smoothly to a special Lagrangian submanifold in the Hamiltonian isotopy class of S.

Exact Lagrangians in plumbings

Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Spin Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A2-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.

Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangian submanifolds

We establish a new version of Floer homology for monotone Lagrangian embeddings in symplectic manifolds. As applications, we get assertions for monotone Lagrangian submanifolds L\hookrightarrow M which are displaceable through Hamiltonian isotopies (this happens for instance when M=\mathbb{C}^{n} ). We show that when L is aspherical, or more generally when the homology of its universal cover vanishes in odd degrees, its Maslov number N_{L} equals 2. This is a generalization of Audin’s conjecture. We also give topological characterisations of Lagrangians L\hookrightarrow M with maximal Maslov number: when N_{L}=\dim (L)+1 then L is homeomorphic to a sphere; when N_{L}=n\geq 6 then L fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in T^{\ast}S^{2k+1} whose Maslov class is zero is homeomorphic to S^{2k+1} .

Nearby Lagrangians with vanishing Maslov class are homotopy equivalent

We prove that the inclusion of every closed exact Lagrangian with vanishing Maslov class in a cotangent bundle is a homotopy equivalence. We start by adapting an idea of Fukaya-Seidel-Smith to prove that such a Lagrangian is equivalent to the zero section in the Fukaya category with integral coefficients. We then study an extension of the Fukaya category in which Lagrangians equipped with local systems of arbitrary dimension are admitted as objects, and prove that this extension is generated, in the appropriate sense, by local systems over a cotangent fibre. Whenever the cotangent bundle is simply connected, this generation statement is used to prove that every closed exact Lagrangian of vanishing Maslov index is simply connected. Finally, we borrow ideas from coarse geometry to develop a Fukaya category associated to the universal cover, allowing us to prove the result in the general case.

On Maslov class rigidity for coisotropic submanifolds

We define the Maslov index of a loop tangent to the characteristic foliation of a coisotropic submanifold as the mean Conley-Zehnder index of a path in the group of linear symplectic transformations, incorporating the rotation of the tangent space of the leaf—this is the standard Lagrangian counterpart—and the holonomy of the characteristic foliation. We also show that, with this definition, the Maslov class rigidity extends to the class of the so-called stable coisotropic submanifolds including Lagrangian tori and stable hypersurfaces.