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