Related papers: Lagrangian caps

We consider deformations of singular Lagrangian varieties in symplectic spaces. We show the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations. Using this result, we prove that, under some assumptions, a…

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,…

Given a sequence of genus $g\geq 2$ curves converging to a punctured Riemann surface with complete metric of constant Gaussian curvature $-1$. we prove that the Kodaira embedding using orthonormal basis of the Bergman space of sections of a…

We prove a quantitative $h$-principle statement for subcritical isotropic embeddings. As an application, we construct a symplectic homeomorphism that takes a symplectic disc into an isotropic one in dimension at least $6$.

In this note we provide explicit constructions of exact Lagrangian embeddings of tori and Klein bottles inside the symplectisation of an overtwisted contact three-manifold. Note that any closed exact Lagrangian in the symplectisation is…

Given a class of embeddings into a contact or a symplectic manifold, we give a sufficient condition, that we call isocontact or isosymplectic realization, for this class to satisfy a general $h$-principle. The flexibility follows from the…

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On…

We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value…

In this short note we observe that the boundary of a properly embedded compact exact Lagrangian sub-manifolds in a subcritical Weinstein domain $X$ necessarily admits Reeb chords. The existence of this Reeb chords either follows from an…

We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary.

We prove two results on the classification of trivial Legendrian embeddings $g: G \rightarrow (S^3,\xi_{std})$ of planar graphs. First, the oriented Legendrian ribbon $R_g$ and rotation invariant $\text{rot}_g$ are a complete set of…

The space Loc(m,S) of rank m flat bundles on a closed surface S is K_2-symplectic. A threefold M bounding S gives rise a K_2-Lagrangian in Loc(m,S) given by the flat bundles on S extending to M. We generalize this, replacing the zero…

We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian…

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,\omega)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal…

We construct a family of uncountably many Lagrangian submanifolds in the standard bidisks such that the Lagrangian Hofer diameter associated to each Lagrangian submanifold is unbounded. We also prove a certain inequality of the Lagrangian…

This paper studies linear relations and their self-adjoint realizations arising from 2d-dimensional canonical systems, with a focus on how the symplectic structure interacts with boundary conditions. Understanding this interplay allows us…

We construct three sequences of regular surfaces of general type with unbounded numerical invariants whose canonical map is 2-to-1 onto a canonically embedded surface. Only sporadic examples of surfaces with these properties were previously…

We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m-dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6,s if the…

Let M be a compact symplectic manifold, and L be a closed Lagrangian submanifold which can be lifted to a Legendrian submanifold in the contactization of M. For any Legendrian deformation of L satisfying some given conditions, we get a new…

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K…