Monotone Lagrangian Submanifolds Research Articles

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].

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.

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.

The wall-crossing formula and Lagrangian mutations

We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behaviour away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one.In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds. In higher dimension, this provides new Lagrangian tori in toric Fanos corresponding to different chambers of the mirror variety, including ones which are conjecturally separated by infinitely many walls from the chamber containing the standard toric fibre.

Symplectic topology of Lagrangian submanifolds of ℂPn with intermediate minimal Maslov numbers

Abstract We examine symplectic topological features of a certain family of monotone Lagrangian submanifolds in ℂPn . First we give cohomological constraints on a Lagrangian submanifold in ℂPn whose first integral homology is p-torsion. In the case where (n, p) = (5,3), (8, 3), we prove that the cohomologies with coefficients in ℤ2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3)ℤ3) and SU(3)/ℤ3, respectively. Then we calculate the Floer cohomology with coefficients in ℤ2 of a monotone Lagrangian submanifold SU(p)/ℤ p in C P p 2 − 1 . ${\mathbb C}P^{p^2-1}.$

On certain Lagrangian submanifolds of S2×S2 and ℂPn

We consider various constructions of monotone Lagrangian submanifolds of $C P^n, S^2\times S^2$, and quadric hypersurfaces of $C P^n$. In $S^2\times S^2$ and $C P^2$ we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of $C P^2$ is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadrics and of $C P^n$ which can be understood either in terms of the geodesic flow on $T^*S^n$ or in terms of the Biran circle bundle construction. Unlike previously-known monotone Lagrangian submanifolds of closed simply connected symplectic manifolds, many of our higher-dimensional Lagrangian submanifolds are provably displaceable.

On the topology of monotone Lagrangian submanifolds

We find new obstructions on the topology of monotone Lagrangian submanifolds of $C^{n}$ under some hypothesis on the homology of their universal cover. In particular we show that nontrivial connected sums of manifolds of odd dimensions do not admit monotone Lagrangian embeddings into $\C^{n}$ whereas some of these examples are known to admit usual Lagrangian embeddings. In dimension three we get as a corollary that the only orientable Lagrangians in ${\bf C}^{3}$ are products $S^{1}\times \Sigma$.

Unlinking and unknottedness of monotone Lagrangian submanifolds

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

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} .

Rigidity and uniruling for Lagrangian submanifolds

This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.

Rigid subsets of symplectic manifolds

AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Mean curvature flow of monotone Lagrangian submanifolds

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $${\mathbb{c}}^n$$ .

The ℤ–graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds

We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth formula for the spectral sequence and an ring structure on it. The ring structure on the Z_Sigma(L)-graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian sub-manifold via the spectral sequence. Using the Z-graded symplectic Floer cohomology, we show some intertwining relations among the Hofer energy e_H(L) of the embedded Lagrangian, the minimal symplectic action sigma(L), the minimal Maslov index Sigma(L)$ and the smallest integer k(L, phi$ of the converging spectral sequence of the Lagrangian L.

Some examples of Lagrangian submanifolds of the moduli space of flat connections

We will construct monotone Lagrangian submanifolds of a well-known finite ramified covering of the moduli space of flat connections on a nontrivial $SO(3)$-bundle over a closed oriented surface. Moreover we will give some examples of the Floer cohomology group of Lagrangian submanifolds.