Related papers: Infinitely many monotone Lagrangian tori in higher…

We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and…

For a broad class of symplectic manifolds of dimension at least six, we find the following new phenomenon: there exist local exotic Lagrangian tori. More specifically, let $X$ be a geometrically bounded symplectic manifold of dimension at…

We construct infinitely many families of monotone Lagrangian tori in $\mathbb{R}^6$, no two of which are related by Hamiltonian isotopies (or symplectomorphisms). These families are distinguished by the (arbitrarily large) numbers of…

We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to…

We show the existence of special Lagrangian tori on one family of Borcea-Voisin threefolds. We also construct a family of special Lagrangian submanifolds on the total space of the canonical line bundle of projective spaces.

This small note contains two simple remarks about Bohr - Sommerfeld with respect to the anticanonical class lagrangian tori in the projective spaces, which are the most popular examples of monotone symplectic manifolds.

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get…

We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We…

In this paper we show that there exist simply connected symplectic 4-manifolds which contain infinitely many knotted lagrangian tori, i.e. lagrangian embeddings of tori that are homotopic but not isotopic. Moreover, the homology class they…

Twist tori are examples of exotic monotone lagrangian tori, presented in [1]. This tree of examples grew up over the first one --- the torus $\Theta \in \R^4$, constructured in [2] and [3]. On the other hand, in [4] and [5] we proposed a…

We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behavior away from a Donaldson divisor. We define geometric operations called mutations of…

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone…

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…

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…

We study special Lagrangian cones in $\C^n$ with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in $\C^3$ each of which has a toroidal link. We obtain a detailed geometric description of…

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…

In recent papers, summarized in survey [1], we construct a number of examples of non standard lagrangian tori on compact toric varieties and as well on certain non toric varieties which admit pseudotoric structures. Using this pseudotoric…

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 describe an iterative construction of Lagrangian tori in the complex Grassmannian $\operatorname{Gr}(k,n)$, based on the cluster algebra structure of the coordinate ring of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. Each…

We construct an exotic monotone Lagrangian torus in CP^2 using techniques motivated by mirror symmetry. We show that it bounds 10 families of Maslov index 2 holomorphic discs, and it follows that this exotic torus is not Hamiltonian…