Related papers: Conformal Structures and Necksizes of Embedded Con…

We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space $\Mk$ of…

We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method…

The aim of this paper is to extend classic results of the theory of CMC surfaces in the product spaces to the class of immersed surfaces in $\mathbb{M}^2(\kappa)\times\mathbb{R}$ whose mean curvature is given as a $C^1$ function depending…

It is known that complex constant mean curvature ({\sc CMC} for short) immersions in $\mathbb C^3$ are natural complexifications of {\sc CMC}-immersions in $\mathbb R^3$. In this paper, conversely we consider {\it real form surfaces} of a…

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The…

We prove uniform convergence of metrics $g_k$ on a closed surface with bounded integral curvature (measure) in the sense of A.D. Alexandrov, under the assumption that the curvature measures $\mathbb{K}_{g_k}=\mu^1_k-\mu^2_k$, where…

We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in…

We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the…

Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $\Sigma$ be…

We prove that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits a proper conformal constant mean curvature one ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space…

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the…

The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural real structure i.e. it admits an anti-holomorphic involution $\sigma$. The involution…

Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh, Ketover and Maximo we show that in any closed Riemannian 3-manifold $(M,g)$, the genus of an embedded CMC surface can be bounded only in…

We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These…

We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the 3-dimensional hyperbolic space $\mathbb{H}^3$. We go further and establish a jet interpolation theorem…

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $\Sigma$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent…

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…

Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let…

We start the investigation of immersions $\Psi$ of a simply connected domain $D$ into three dimensional Euclidean space $R^3$, which have constant mean curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which leave…

One of the basic geometric objects in conformal field theory (CFT) is the the moduli space of Riemann surfaces whose $n$ boundaries are ''rigged'' with analytic parametrizations. The fundamental operation is the sewing of such surfaces…