Related papers: Mergelyan approximation theorem for holomorphic Le…

In this paper, we prove that every confomal minimal immersion of an open Riemann surface into $\mathbb{R}^n$ for $n\ge 5$ can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open…

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…

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by…

Let M be a compact Sasakian manifold. We show that M admits a CR-embedding into a Sasakian manifold diffeomorphic to a sphere, and this embedding is compatible with the respective Reeb fields. We argue that a stronger embedding theorem…

This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces…

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…

Y. Eliashberg and N. Mishachev introduced the notion of wrinkled embedding to show that any tangential homotopy can be approximated by a homotopy of topological embeddings with mild singularities. This concept plays an important role in…

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all…

In this paper we show that the space of holomorphic immersions from any given open Riemann surface, $M$, into the Riemann sphere $\mathbb{CP}^1$ is weakly homotopy equivalent to the space of continuous maps from $M$ to the complement of the…

We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured…

Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective…

The purpose of this paper is to prove a gluing theorem for a given special Lagrangian submanifold of a Calabi-Yau 3-fold. The proof will be an adaption of the gluing techniques in J-holomorphic curve theory. It is a well known procedure in…

The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan…

We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields…

In this paper we prove that every bordered Riemann surface M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean $3$-space $\mathbb{C}^3$. The real part of such an embedding is a complete conformal…

We construct a complete, bounded Legendrian immersion in C^3. As direct applications of it, we show the first examples of a weakly complete bounded flat front in hyperbolic 3-space, a weakly complete bounded flat front in de Sitter 3-space,…

Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $\Lambda\subset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $\Lambda$ we get a Weinstein cobordism $X_{\Lambda}$ with a…

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact…

In any 5 dimensional closed Sasakian manifold we prove that any minmax operation on the area among Legendrian surfaces is achieved by a continuous conformal Legendrian map from a closed riemann surface $S$ into $N^5$ equipped with an…

We prove a compactness theorem for holomorphic curves in 4-dimensional symplectizations that have embedded projections to the underlying 3-manifold. It strengthens the cylindrical case of the SFT compactness theorem by using intersection…