Consider a conformal immersion $f: B(0,1)\setminus \{0\}\subset \mathbb{R}^2 \to \mathbb{R}^3$. Let $g$ be the first fundamental form of $f(B(0,1)\setminus \{0\})\subset \mathbb{R}^3$. Assume that its second fundamental form $A$ satisfies $$ \int_{B(0,1)} |A|^2_g d\mathrm{vol}_g <\infty. $$ Can we find a family of smooth immersions $(f_\delta)_{\delta\in(0,1)}$ satisfying the following three conditions ?
- Each $f_\delta$ is a smooth immersion on the whole ball $B(0,1)$,
- we have the limit $f_\delta \xrightarrow[\delta\to 0]{}{f}$ in the $C^k_{loc}$-topology on $B(0,1)\setminus \{0\}$ for any $k\in\mathbb{N}$,
- we have a $L^2$-bound on the second fundamental forms : if $g_\delta$ and $A_\delta$ are respectively the first and second fundamental form of $f_\delta$, we ask that $$ \sup_{\delta\in(0,1)} \int_{B(0,1)} |A_\delta|^2_{g_\delta} d\mathrm{vol}_{g_\delta} <\infty. $$