$r<s$, the unit circle of ${\mathbb{C}}^*$ is mapped onto a parallel circle of minimal radius, called a “neck” of the Delaunay surface. As $t$ tends to $0$, the immersions degenerate into a point on every compact subset of ${\mathbb{C}}^*$. Nevertheless, we call this family the “catenoidal family” because applying a blowup to the immersions makes them converge towards a catenoidal immersion of ${\mathbb{R}}^3$ on every compact subset of ${\mathbb{C}}^*$ (see Section \[sectionBlowUp\] for more details). In both cases, the weight of the induced surfaces is given by $w = 2\pi t$.

Perturbed Delaunay immersions {#sectionPerturbedDelaunayImmersions}
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In this section, we study the immersions induced by a perturbation of Delaunay DPW data with small non-vanishing weights in a neighbourhood of $z=0$. Our results are the same wether we choose the spherical or the catenoidal family of immersions. We thus drop the index $r,s$ in the Delaunay DPW data and replace it by a small value of $t=4rs\sinh q$ in a neighbourhood of $t=0$ such that $$t<T_{\max}:=\frac{\tanh \frac{q}{2}}{2}.$$ For all $\epsilon>0$, we denote $$D_\epsilon := \left\{ z\in{\mathbb{C}}\mid |z|<\epsilon \right\}, \qquad  D_\epsilon^* := D_\epsilon \backslash\left\{0\right\}.$$

\[defPerturbedDelaunayPotential\] Let $\rho>e^q$, $0<T<T_{\max}$ and $\epsilon>0$. A perturbed Delaunay potential is a continuous one-parameter family $(\xi_t)_{t\in (-T,T)}$ of 