Related papers: Remarks on monotone Lagrangians in $\mathbf{C}^n$

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum…

We provide a proof that the $SU(2)$ character variety of a genus two surface, $\chi(F_2)$, is a closed compact manifold, and a proof of the Narasimhan-Ramanan theorem that $\chi(F_2)$ is homeomorphic to ${\mathbb C} P^3$. This is done…

We prove that there is an algorithm which determines whether or not a given 2-polyhedron can be embedded into some integral homology 3-sphere. This is a corollary of the following main result. Let $M$ be a compact connected orientable…

We define new Hamiltonian isotopy invariants for a monotone Lagrangian torus embedded in a symplectic 4-manifold. We show that, in the standard symplectic 4-space, these invariants distinguish a monotone Clifford torus from a Chekanov…

In this note we define the notion of collarable slices of Lagrangian submanifolds. Those are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact…

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not…

We study the topological entropy of the Lagrangian flow restricted to an energy level $E_{L}^{-1}(c) \subset TM$ for $ c >e_0(L)$. We prove that if the flow of the Tonelli Lagrangian $ L: M \to \mathbb{R}$, on a closed manifold of dimension…

We prove the existence of a subclass of overtwisted contact structures, called strongly overtwisted, on a 3-manifold that satisfy a complete h-principle without prescribing the contact structures over any subset of the 3-manifold. As a…

Let $L_0,L_1,L_2 \subset M$ be exact Lagrangian spheres in a Liouville domain $M$ with $2c_1(M)=0$. If $L_0,L_1,L_2$ form an $A_3$-configuration, we show that $\mathscr{L}(L_0)$ and $\mathscr{L}(L_2)$ endowed with the Hofer metric contain…

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…

In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of…

Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a…

An isometric immersion $f:M^n\to \tilde M^n$ from a Riemannian $n$-manifold $M^n$ into a K\"ahler $n$-manifold $\tilde M^n$ is called {\it Lagrangian} if the complex structure $J$ of the ambient manifold $\tilde M^n$ interchanges each…

This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally…

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression…

We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and $\gamma$-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each…

Every Maslov-zero Lagrangian torus in a K3 surface has non-trivial homology class. This note aims to extend this result to Lagrangian tori with Maslov indices congruent to zero modulo 4. Conversely, we show that every homologically…

Given two locally conformal symplectic (LCS) structures on manifolds $M_1$ and $M_2$, we construct a natural $\R^+$-torsor of locally conformal symplectic structures on a certain covering space $M_1 \boxplus M_2$ of $M_1 \times M_2$. As the…

The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of…

In this paper, we prove that there exists a universal constant $C$, depending only on positive integers $n\geq 3$ and $p\leq n-1$, such that if $M^n$ is a compact free boundary submanifold of dimension $n$ immersed in the Euclidean unit…