Hi, I would like to know if it exists a sequence of $C^2$ immersion $f_k: S^2 \rightarrow \mathbb{R}^3$ which converge (in C^2) to $z^2$ except on a finite set of point, i.e $f^k \rightarrow z^2$ in $C^2_{loc}(S^2\setminus \{ a_1, \dots , a_n \})$.

Here $S^2$ is identified to $\hat{\mathbb{C}}$ the Riemann sphere, hence $z^2: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}} \sim S^2 \subset \mathbb{R}^3$ makes sense. In fact my question is about $P/Q$ where $P$ and $Q$ are two element of $\mathbb{C}[z]$, but we can start with $z^2$ in order to make it more clear.

It looks very hard topologically. For instance if I assume "embedded" instead of "immersed" it is not very difficult to prove that such a sequence doesn't exist. But I can't show more, especially, would like to know if

1) it exists, assuming we have have a sequence of immersion from a ball to $\mathbb{R}^3$ which satisfies the same hypothesis on the boundary or if the curvature if bounded from above?

2) how to produce such a sequence in the general case? In fact looking at a proof it seems to looks like something like the sphere eversion: no topological obstruction but no way to see the effective map.

I hope to be clear, Thanks in advance for your contribution.