Related papers: Homological Lagrangian monodromy and Wang exact se…

Let $M$ be a Fano manifold equipped with a K\"ahler 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…

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of…

Let $(X, \omega)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $\mu$. Take an invariant $\omega$-compatible almost complex structure, we consider…

We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of…

The endpoint of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. In the first paper, we…

We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology class of a displaceable monotone…

For any irreducible compact homogeneous K\"ahler manifold, we classify the compact tight Lagrangian submanifolds which have the Z_2-homology of a sphere.

This short and fairly informal note is an attempt to explain how methods of homological algebra may be brought to bear on problems in symplectic geometry. We do this by looking at a familiar sample question, which is that of the topology of…

We show that an exact Lagrangian cobordism $L\subset \mathbb R \times P \times \mathbb R$ from a Legendrian submanifold $\Lambda\subset P\times \mathbb R$ to itself satisfies $H_i(L;\mathbb F)=H_i(\Lambda;\mathbb F)$ for any field $\mathbb…

In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary…

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point…

Let X be the Fermat quintic threefold. The set of real solutions L forms a Lagrangian submanifold of X. Multiplying the homogeneous coordinates of X by various fifth roots of unity gives automorphisms of X; the images of L under these…

Following an idea of Fr\'ed\'eric le Roux, we define in this paper a family of Hofer-type pseudonorms on braid groups, computing the minimal energy of a Hamiltonian diffeomorphism which fixes a Lagrangian configuration of circles on the…

We establish a cosymplectic counterpart of Banyaga's theorem by proving that the group of weakly Hamiltonian diffeomorphisms, $\Ham_{\eta,\omega}(M)$, is simple on any closed cosymplectic manifold. A key structural result, derived from Lie…

We consider paths of Hamiltonian diffeomorphism preserving a given compact monotone Lagrangian in a symplectic manifold that extend to an $S^1$--Hamiltonian action. We compute the leading term of the associated Lagrangian Seidel element. We…

For an adiscal or monotone regular coisotropic submanifold $N$ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $N$. Given a Hamiltonian isotopy $\phi=(\phi^t)$ and a…

We establish a criterion on wrapped Floer homology of an exact Lagrangian sub- manifold in a Liouville domain, which ensures the almost-existence of Hamiltonian chords near a given energy level. To this purpose we introduce a relative…

The purpose of this short note is to prove the uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to RP^n in CP^n modulo isometric group actions.

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ admits holomorphic connections with…

We study the monodromy representations underlying compact Lagrangian fibrations. In the case where the associated period map is generically immersive, we prove that the mondromy representation is irreducible over $\mathbb{C}$. In the…