Related papers: Polyhedral approximation by Lagrangian and isotrop…

We prove a flat torus theorem for quadric complexes. In particular, we show that if a non-cyclic free abelian group $G$ acts metrically properly on a quadric complex $X$, then $G \cong \mathbb{Z}^2$ and $X$ contains a $G$-invariant…

We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.

We propose a new method for the construction of Hamiltonian-minimal and minimal Lagrangian immersions of some manifolds in $C^n$ and in $CP^n$. By this method one can construct, in particular, immersions of such manifolds as the generalized…

We construct a family of Lagrangian tori $\Theta^n_s$ $\subset$ $(\mathbb{C}P^1)^n$, $s \in (0,1)$, where $\Theta^n_{1/2} = \Theta^n$, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for $n = 2m$ and $s…

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and spaces with mixed logarithmic smoothness. Equivalent norms of a space with mixed logarithmic smoothness are found and…

We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in…

We prove that any asymptotically locally Euclidean scalar-flat K\"ahler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector…

In this remark we discuss a relationship between (co)homology classes of a symplectic manifold realized by symplectic and lagrangian objects. We establish some transversality condition for the classes, realized by symplectic divisors and…

Let k > 2. We show that if a closed orientable 2k-manifold K, with Euler characteristic not equal to -2, admits an exact Lagrangian immersion into complex Euclidean 2k-space with one transverse double point and no other self-intersections,…

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…

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…

The Floer homology of (RP^n,T^n) is calculated, for n odd.

We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy…

In recent years methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this note it is shown that the latter method is actually…

We show that the Lagrangian torus in the cotangent bundles of the 2-sphere obtained by applying the geodesic flow to the unit circle in a fibre is not displaceable by computing its Lagrangian Floer homology. The computation is based on a…

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can…

We investigate the possibility of embedding minimal abelian surfaces in smooth toric 4-folds with Picard number 2. The existence of such an embedding imposes conditions on the 4-fold, which we partly describe. On the other hand, we exhibit…

For each rational number $p/q\in (1/2,\sqrt 2/2)$ one can construct an $\mathbb S^1$-equivariant minimal torus in $\mathbb S^3$ called Otsuki torus and denoted by $O_{p/q}$. The Lawson's bipolar surface construction applied to $O_{p/q}$…

We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-\gamma}}$ for $|X|$ large, when $A<0$ and…

Consider a 2-plane $P \subset \mathbb{C}^n$ and let $D$ be a bounded region in $P$ with a piecewise-smooth boundary. Let $I(D)$ be the infimum of areas of all piecewise-smooth isotropic surfaces in $\mathbb{C}^n$ with the same boundary as…