---
abstract: 'Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in hyperbolic space.'
author:
- Thomas Raujouan
title: 'Constant mean curvature $n$-noids in hyperbolic space'
---

Introduction {#introduction .unnumbered}
============

In [@dpw], Dorfmeister, Pedit and Wu introduced a loop group method (the DPW method) for constructing harmonic maps from a Riemann surface into a symmetric space. As a consequence, their method provides a Weierstrass-type representation of constant mean curvature surfaces (CMC) in Euclidean space ${\mathbb{R}}^3$, three-dimensional sphere ${\mathbb{S}}^3$, or hyperbolic space ${\mathbb{H}}^3$. Many examples have been constructed (see for example [@newcmc; @dw; @kkrs; @dik; @heller1; @heller2]). Among them, Traizet [@nnoids; @minoids] showed how the DPW method in ${\mathbb{R}}^3$ can construct genus zero $n$-noids with Delaunay ends (as Kapouleas did with partial differential equations techniques in [@kapouleas]) and glue half-Delaunay ends to minimal surfaces (as did Mazzeo and Pacard in [@pacard], also with PDE techniques). A natural question is wether these constructions can be carried out in ${\mathbb{H}}^3$. Although properly embedded CMC annuli of mean curvature $H>1$ in ${\mathbb{H}}^3$ are well-known since the work of Korevaar, Kusner, Meeks and Solomon [@meeks], no construction similar to [@kapouleas] or 