to $0$, and to estimate the growth of their unitary part around the end at $z=0$. Section \[sectionConvergenceImmersions\] proves that these immersions do converge, which is the first point of Theorem \[theoremPerturbedDelaunay\]. Before proving the embeddedness in Section \[sectionEmbeddedness\], Section \[sectionConvergenceNormales\] is devoted to the convergence of the normal maps. Finally, we compute the limit axes in Section \[sectionLimitAxis\].

A property of $\xi_t$ {#sectionPropertyXit}
---------------------

We begin by diagonalising $A_t$ in a unitary basis (Proposition \[propDiagA\]) in order simplify the computations in Proposition \[propC3\], in which we use the Fröbenius method for a fixed value of ${\lambda}=e^{\pm q}$. This will ensure the existence of the ${\mathcal{C}}^1$ map $P_1\in{\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_\rho}$ that will be used in Section \[sectionzAP\] to define the factor $P$ in the $Mz^AP$ form of $\Phi_t$.

\[propDiagA\] There exist $e^q<R<\rho$ and $0<T'<T$ such that for all $t\in(-T',T')$, $A_t = H_tD_tH_t^{-1}$ with $H_t\in{\Lambda {\mathrm{SU}}(2)_R}$ and $iD_t\in{\Lambda {\mathfrak{su}}(2)_R}$. Moreover, $H_t$ and $D_t$ are smooth with respect to $t$.

For all ${\lambda}\in{\mathbb{S}}^1$, $$\begin{aligned}
\label{eqDetA}
        -\det A_t ({\lambda}) &= \frac{1}{4} + \frac{t{\lambda}^{-1}({\lambda}-e^q)({\lambda}-e^{-q})}{4\sinh q}\\
        &= \frac{1}{4} + \frac{t}{4\sinh q}\left( {\lambda}+{\lambda}^{-1} - 2 \cosh q \right) \in {\mathbb{R}}.\nonumber
    \end{aligned}$$ Extending this determinant as a holomorphic function on ${\mathbb{A}}_{{\rho}}$, there exists $T'>0$ such that $$\left|-\det A_t({\lambda}) 