the catenoidal family ($r<s$)}.$$

Let $\xi_t$ and $\Phi_t$ as in Theorem \[theoremPerturbedDelaunay\] with $\rho$, $T$ and $\epsilon$ fixed. This Section is dedicated to the proof of Theorem \[theoremPerturbedDelaunay\].

The ${\mathcal{C}}^2$-regularity of $\xi_t$ essentially means that $C_t(z)$ is ${\mathcal{C}}^2$ with respect to $(t,z)$. Together with the ${\mathcal{C}}^2$-regularity of $\phi_t$, it implies that $\Phi_t$ is ${\mathcal{C}}^2$ with respect to $(t,z)$. Thus, ${\mathcal{M}}(\Phi_t)$ is also ${\mathcal{C}}^2$ with respect to $t$. These regularities and the fact that there exists a solution $\Phi_t$ solving the monodromy problem are used in Section \[sectionPropertyXit\] to deduce an essential piece of information about the potential $\xi_t$ (Proposition \[propC3\]). This step then allows us to write in Section \[sectionzAP\] the holomorphic frame $\Phi_t$ in a $Mz^AP$ form given by the Fröbenius method (Proposition \[propzAP1\]), and to gauge this expression, in order to gain an order of convergence with respect to $z$ (Proposition \[propzAP2\]). During this process, the holomorphic frame will loose one order of regularity with respect to $t$, which is why Theorem \[theoremPerturbedDelaunay\] asks for a ${\mathcal{C}}^2$-regularity of the data. Section \[sectionDressedDelaunayFrames\] is devoted to the study of dressed Delaunay frames $Mz^A$ in order to ensure that the immersions $f_t$ will converge to Delaunay immersions as $t$ tends 