Related papers: Lagrangian caps
In this paper we study the uniqueness of Lagrangian fillings of the standard Legendrian sphere $\mathcal{L}_0$ in the standard contact sphere $(S^{2n-1}, \xi_{\text st})$. We show that every exact Maslov zero Lagrangian filling $L$ of…
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 there are at least seeds many exact embedded Lagrangian fillings for Legendrian links of type $\mathsf{ADE}$. We also provide seeds many Lagrangian fillings with certain symmetries for type $\mathsf{BCFG}$. Our main tools are…
In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove…
A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+\alpha}$-smooth surfaces. The Conjecture is first…
We prove an existence result for exact lagrangian cobordisms between closed legendrians.
We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same…
We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$,…
Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of…
Let Gr(2, E) be the Grassmann bundle of two-planes associated to a general bundle E over a curve X. We prove that an embedding of Gr(2, E) by a certain twist of the relative Pl\"ucker map is not secant defective. This yields a new and more…
We determine when a Legendrian quasipositive 3-braid closure in standard contact $\mathbb{R}^3$ admits an orientable or non-orientable exact Lagrangian filling. Our main result provides evidence for the orientable fillability conjecture of…
By a result due to Ziltener, there exist no closed embedded Bohr-Sommerfeld Lagrangians inside $\mathbb CP^n$ for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone…
We construct exact Lagrangian fillings of Legendrian torus links $\Lambda(k, n-k)$ that are fixed by a Legendrian loop that acts by $2\pi\ell/n$ rotation. Using these rotationally symmetric fillings, we produce fillings of the corresponding…
We show that a two-dimensional totally real concordance can be approximated by a Lagrangian concordance whose Legendrian boundary has been stabilised both positively and negatively sufficiently many times. The main applications that we…
Let $L$ be a compact oriented Lagrangian surface in a K\"ahler surface endowed with a complete Riemannian metric (compatible with the symplectic structure and the complex structure) with bounded sectional curvatures and a positive lower…
In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a…
We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in $\mathbb{C}^2$. We recall some classical obstructions and prove some deeper ones related to symplectic…
We define a differential graded algebra associated to Legendrian knots in thickened convex surfaces $\Sigma\times \mathbb{R}$. The algebra is defined in the same spirit as the Chekanov-Eliashberg DGA for Legendrians in $\mathbb{R}^3$, but…
It is shown that a Lagrangian inclusion of a real surface in $\mathbb C^2$ with a standard open Whitney umbrella and double transverse self-intersections is rationally convex.
We construct the first smooth embedded compact special Legendrian surfaces in \(\mathbb S^5\) of genus greater than one. More precisely, for every sufficiently large integer \(k\), we construct an embedded special Legendrian surface whose…