Related papers: On an exotic Lagrangian torus in $\mathbb{C}P^2$

We show that any weakly partially hyperbolic diffeomorphism on the 2-torus may be realized as the dynamics on a center-stable or center-unstable torus of a 3-dimensional strongly partially hyperbolic system. We also construct examples of…

We study the Lagrangian description of chiral bosons, p-form gauge fields with (anti-)self-dual gauge field strengths, in D=2p+2 dimensional spacetime of nontrivial topology. We show that the manifestly Lorentz and diffeomorphism invariant…

In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for…

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…

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…

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to…

An elliptic pair $(X, C)$ is a generalization of a rational elliptic fibration $X \to \mathbb{P}^1$ with fiber $C,$ introduced in \cite{jenia_blowup}. Here, $X$ is a projective rational surface with log terminal singularities, and $C$ is an…

We extend a packing result of R. Hind and E. Kerman for integral Lagrangian tori in $\mathbb{S}^{2} \times \mathbb{S}^{2}$ to the Del Pezzo surfaces $(\mathbb{D}_{n}, \omega_{\mathbb{D}_{n}})$ for $n = 1, \dots, 5$. An integral torus is one…

We analyse a simple Chiang-type lagrangian in CP2 which is topologically an RP2 but exhibits a distinguishing behaviour under reduction by one of the toric circle actions, in particular it is a one-to-one transverse lifting of a great…

In this paper, we present an algorithm to computea fiberwise Lagrangian torus in quasi-periodic (QP) Hamiltonian systems, whose convergence is proved in the [CHP25]. We exhibit the algorithm with two models. The first is a Tokamak model…

The model of dual superconductivity has been revisited considering the $U(1)$-gauged Ginzburg-Landau lagrangian density on a non-commutative torus $T^{2}_{NC}$, according to a new approach, we propose, in dealing with non-commutative space…

In an earlier paper, we proved the connectedness of the fibers of every $2n$-dimensional integrable system satisfying both: the action extends the action of an $(n-1)$-dimensional torus which has a proper moment map, and every tall singular…

We adapt and generalize McDuff's method of probes from toric system to so called semitoric integrable systems and apply it to study the (non)displaceability properties of the fibers of $3$ examples of semitoric integrable systems.

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They can be considered as a generalization of special Lagrangians or Lagrangian and minimal…

We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to…

Let $\TT$ be a torus. We prove that all subsets of $\TT$ with finitely many boundary components (none of them being points) embed properly into $\CC^2$.

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…

Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple)…

We 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…

Let $f: \mathbb{T}^3\to\mathbb{T}^3$ be a partially hyperbolic diffeomorphism on the 3-torus $\mathbb{T}^3$. In his thesis, Hammerlindl proved that for lifted center foliation $\mathcal{F}^c_f$, there exists $R>0$, such that for any $x\in…