surfaces $M_t$ have genus zero and $n$ Delaunay ends.

2.  After a suitable blow-up, $M_t$ converges to $M_0$ as $t$ tends to $0$.

3.  If $M_0$ is Alexandrov-embedded, then all the ends of $M_t$ are of unduloidal type if $t>0$ and of nodoidal type if $t<0$. Moreover, $M_t$ is Alexandrov-embedded if $t>0$.

Following the proofs of [@nnoids; @minoids] gives an effective strategy to construct the desired CMC surfaces $M_t$. This is done in Sections \[sectionNnoids\] and \[sectionMinoids\]. However, showing that $M_t$ is Alexandrov-embedded requires a precise knowledge of its ends. This is the purpose of the main theorem (Section \[sectionPerturbedDelaunayImmersions\], Theorem \[theoremPerturbedDelaunay\]). We consider a family of holomorphic perturbations of the data giving rise via the DPW method to a half-Delaunay embedding $f_0: {\mathbb{D}}^*\subset {\mathbb{C}}\longrightarrow {\mathbb{H}}^3$ and show that the perturbed induced surfaces $f_t({\mathbb{D}}^*)$ are also embedded. Note that the domain on which the perturbed immersions are defined does not depend on the parameter $t$, which is stronger than $f_t$ having an embedded end, and is critical for showing that the surfaces $M_t$ are Alexandrov-embedded. The essential hypothesis on the perturbations is that they do not occasion a period problem on the domain ${\mathbb{D}}^*$ (which is not simply connected). The 