Related papers: Packing Lagrangian tori

We prove some results about linking of Lagrangian tori in the symplectic vector space $(\mathbb{R}^4, \omega)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for…

In this paper we obtain sharp obstructions to the symplectic embedding of the lagrangian bidisk into four-dimensional balls, ellipsoids and symplectic polydisks. We prove, in fact, that the interior of the lagrangian bidisk is…

We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three…

We give the first examples of Fano manifolds with multiple optimal tori, i.e.~we construct monotone Lagrangian tori $L$, such that the weighted number of holomorphic Maslov index two discs with boundary on $L$ equals the upper bound given…

We show that for $n\geq 2$ there exists an exact Lagrangian submanifold $L$ in the cotangent bundle $T^*\mathbb{T}^n$ of the $n$-dimensional torus $\mathbb{T}^n$ such that $L$ is symplectically but not Hamiltonian isotopic to the zero…

In this article we study covering spaces of symplectic toric orbifolds and symplectic toric orbifold bundles. In particular, we show that all symplectic toric orbifold coverings are quotients of some symplectic toric orbifold by a finite…

An integral product Lagrangian torus in the standard symplectic $\mathbb{C}^2$ is defined to be a subset $\{ \pi|z_1|^2 = k, \, \pi|z_2|^2 =l \}$ with $k,l \in \mathbb{N}$. Let $\mathcal{L}$ be the union of all integral product Lagrangian…

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant…

We prove that for any compact orientable connected 3-manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open 2-disk removed admits a Lagrangian embedding into the standard…

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real…

The shape invariant of a symplectic manifold encodes the possible area classes of embedded Lagrangian tori. Potentially this is a powerful invariant, but for most manifolds the shape is unknown. We compute the shape for 4 dimensional…

We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone $S^2\times S^2$, namely any real Lagrangian torus in $S^2\times S^2$ is Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$. The proof is based…

We show that the space of Lagrangians which are Hamiltonian isotopic to the Clifford torus in a complex projective space or in the four-dimensional quadric, taken with Chekanov's Lagrangian Hofer metric, contains a quasi-isometric copy of…

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits…

We define an equivariant Lagrangian Floer theory for Lagrangian torus fibers in a compact symplectic toric manifold equipped with a subtorus action. We show that the set of all Lagrangian torus fibers with weak bounding cochain data whose…

Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is…

We prove that every smoothly immersed 2-torus of $\mathbb{R}^4$ can be approximated, in the C0-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the…

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…

Motivated by work of the first author, this paper studies symplectic embedding problems of lagrangian products that are sufficiently symmetric. In general, lagrangian products arise naturally in the study of billiards. The main result of…

We consider a fibered Lagrangian $L$ in a compact symplectic fibration with small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks with Lagrangian boundary from the base to the total space. In case $L$ is a product,…