Let $E=\mathbb C/\Lambda$ be an elliptic curve,
and let $D\subset E$ be a very small disc.  

($D$ is round for the usual flat metric on $E$)


By the main result of [1], there exists a holomorphic immersion $f:E\setminus D \to \mathbb C$.  
 The image $f(\partial D)$ is a closed curve in $\mathbb C$ that self-intersects a bunch of times.



> 
> What is the shape of $f(\partial D)$?
> 
> 
> 


I understand that such a curve is not unique.
What I want is a qualitative description of an example of such a curve. A drawing would be great.


**Example:**
When $D\subset E$ is a "rather big" disk, the curve ⌘ is the image of $\partial D$ under an immersion $E\setminus D \to \mathbb C\mathbb P^1$ (points in the central square have two preimages; points in the four lobes have zero preimages). But that only works when $D$ is a rather big compared to the size of $E$, and I don't know how to modify this curve as the size of $D$ tends to zero.




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[1]: Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967).



Image of boundary circle under map from punctured elliptic curve to ℂ
I don't know if this is the kind of answer which will satisfy. Write $f$ for the function on $\mathbb{C} - \Lambda$ and $z$ for the coordinate on $\mathbb{C}$. Write $D^{\ast}$ for the punctured disc.


[Forstneric and Ohsawa](https://arxiv.org/abs/1106.0936) show we can take $f$ to be of finite order as $z \to 0$, meaning $|f| < \exp(z^{-N})$ for some $N$. Put $f' = g$, so $g$ is a nowhere vanishing function on $D^{\ast}$ and also of finite order. Let $k$ be the winding number of $g'$ as we travel around $0$, so $z^{-k} g$ has a logarithm on $D^{\ast}$, say $g = z^k e^{h(z)}$. Then $|h| < z^{-N}$ so $h$ is meromorphic. Say $h(z) = c z^{-n} + \cdots$. 


Undoing the changes of variable, $f = \int z^k \exp(c z^{-n} + \cdots)$. 


This integral has different behavior on the sectors where $c z^{-n}$ lies in the left or the right half plane. If we fix $\theta \in (0, \pi/2)$ and consider the $n$ wedges where $\arg(c)-n \arg(z) \in (-\theta, \theta)$, then we have an asymptotic series starting $- \tfrac{z^{k+2n+1}}{c (n+1)} \exp(c z^{-n})$. So, for $z = r e^{i \phi}$ with $\phi$ ranging through each of these wedges, $f$ swings around a circle of large radius, roughly $r^{-n}$ times.


Meanwhile, if $\arg(c)-n \arg(z) \in (\pi-\theta, \pi+\theta)$, then $f'$ is very small, so $f$ stays very near some limiting value. I don't know whether it is the same limiting value in the $n$ different wedges, I don't see why it should be. It circles that value roughly $r^{-n}$ times. However, I am not claiming that I understand the behavior near the $2n$ rays where $\arg(c)-n \arg(z) \in \{0, \pi \}$ well enough to subtract off these numbers of criclings from each other.


