proof relies on the Fröbenius method for linear differential systems with regular singular points. Although this idea has been used in ${\mathbb{R}}^3$ by Kilian, Rossman, Schmitt [@krs] and [@raujouan], the case of ${\mathbb{H}}^3$ generates two extra resonance points that are unavoidable and make their results inapplicable. Our solution is to extend the Fröbenius method to loop-group-valued differential systems.

![*Theorem \[theoremConstructionNnoids\] ensures the existence of $n$-noids with small necks. For $H>1$ small enough ($H\simeq1.5$ on the picture), there exists embedded $n$-noids with more than six ends.*](7noidmainlevee.png){width="7cm"}

Delaunay surfaces in ${\mathbb{H}}^3$ via the DPW method {#sectionNotations}
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This Section fixes the notation and recalls the DPW method in ${\mathbb{H}}^3$.

Hyperbolic space
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#### Matrix model.

Let ${\mathbb{R}}^{1,3}$ denote the space ${\mathbb{R}}^4$ with the Lorentzian metric ${\left\langle x, x \right\rangle} = -x_0^2+x_1^2+x_2^2+x_3^2$. Hyperbolic space is the subset ${\mathbb{H}}^3 $ of vectors $x\in{\mathbb{R}}^{1,3}$ such that ${\left\langle x, x \right\rangle} = -1$ and $x_0>0$, with the metric induced by ${\mathbb{R}}^{1,3}$. The DPW method constructs CMC immersions into a matrix model of ${\mathbb{H}}^3$. Consider the identification $$x=(x_0,x_1,x_2,x_3)\in{\mathbb{R}}^{1,3} \simeq X = \begin{pmatrix}
x_0+x_3 & x_1+ix_2\\
x_1-ix_2 & x_0 - x_3
\end{pmatrix}\in {\mathcal{H}}_2$$ where ${\mathcal{H}}_2:=\{M\in {\mathcal{M}}(2,{\mathbb{C}})\mid M^*=M \}$ denotes the Hermitian matrices. In this model, ${\left\langle X, X 