Related papers: Complete CMC-1 surfaces in hyperbolic space with a…

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 $M$ be an open Riemann surface and let $\Lambda\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $\Lambda$ and…

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct…

Motivated by the work of Bryant on constant mean curvature (CMC) $1$-immersions of surfaces into the hyperbolic space H^3 and after the results of Tarantello (2023), we pursue a possible parametrization for the moduli space of (CMC)…

In this paper, we describe a family of embedded hypersurfaces with constant mean curvature (CMC) in the $(n+1)$-dimensional unit sphere. In the process, we provide evidence for new CMC embedded examples. In particular, for some examples…

We study the geometry of complete immersed surfaces in $\mathbb{R}^3$ with constant anisotropic mean curvature (CAMC). Assuming that the anisotropic functional is uniformly elliptic, we prove that: (1) planes and CAMC cylinders are the only…

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal orbits…

In this paper we investigate the intersection problem for $1$-surfaces immersed in a complete Riemannian three-manifold $P$ with Ricci curvature bounded from below by $-2$. We first prove a Frankel's type theorem for $1$-surfaces with…

We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends…

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and…

We show some area estimates for stable CMC hypersurfaces immersed in Riemannian manifolds with scalar and sectional curvature bounded from below. In particular, we focus on immersions in three-dimensional Riemannian manifolds. As an…

For all open Riemann surface M and real number $\theta \in (0,\pi/4),$ we construct a conformal minimal immersion $X=(X_1,X_2,X_3):M \to \mathbb{R}^3$ such that $X_3+\tan(\theta) |X_1|:M \to \mathbb{R}$ is positive and proper. Furthermore,…

Let $M\subset\mathbb{R}^3$ be a properly embedded, connected, complete surface with boundary a convex planar curve $C$, satisfying an elliptic equation $H=f(H^2-K)$, where $H$ and $K$ are the mean and the Gauss curvature respectively -…

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove…

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface…

Let $H^4$ denote the hyperbolic four-space. Given a bordered Riemann surface, $M$, we prove that every smooth conformal superminimal immersion $\overline M\to H^4$ can be approximated uniformly on compacts in $M$ by proper conformal…

In this paper, firstly, we show the existence of a compact embedded constant mean curvature (CMC) hypersurface $\Sigma_1$ in $\mathbb{S}^{2n}$ of the type $S^{n-1} \times S^{n-1} \times S^{1}$. Moreover, the hypersurface $\Sigma_1$ exhibits…

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k}…

We prove a parametric h-principle for complete nonflat conformal minimal immersions of an open Riemann surface $M$ into $\mathbb R^n$, $n\geq 3$. It follows that the inclusion of the space of such immersions into the space of all nonflat…

We prove that if $X:M^n\to\mathbb{H}^n\times \mathbb{R}$, $n\geq 3$, is a an orientable, complete immersion with finite strong total curvature, then $X$ is proper and $M$ is diffeomorphic to a compact manifold $\bar M$ minus a finite number…