TITLE: Image of boundary circle under map from punctured elliptic curve to ℂ
QUESTION [8 upvotes]: 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.

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

REPLY [2 votes]: Based on David Speyer's answer, I got what I want, namely some intuition about what these closed paths look like.
It's a bit hard to draw them, so I include an inline description.

Let $X:=3i$, $Y:=0$, $Z:=-3i$.
Given $T\in[0,\infty)$, I'll define a closed path in $\mathbb C$, which I'll call $\gamma_T$.
Let $R:=5e^T$ and $r:=e^{-T}$.
Let $C_X$ be the circle of radius $r$ centred around $X$.
Let $C_{Y}$ be the circle of radius $r$ centred around $Y$.
Let $C_{Z}$ be the circle of radius $r$ centred around $Z$.
Let $C_\infty$ be the circle of radius $R$ centred around $0$.
The path $\gamma_T$ is as follows:

• Start at $Re^{-iT}$
  • straight line to $X+re^{-iT}$
  • run along $C_X$ for an angle of $4\pi+2T$, until the point $X+re^{iT}$
  • straight line to $Re^{iT}$
  • run along $C_\infty$ for an angle of $-2T$, until the point $Re^{-iT}$
  • straight line to $Y+re^{-iT}$
  • run along $C_Y$ for an angle of $4\pi+2T$, until the point $Y+re^{iT}$
  • straight line to $Re^{iT}$
  • run along $C_\infty$ for an angle of $-2T$, until the point $Re^{-iT}$
  • straight line to $Z+re^{-iT}$
  • run along $C_Z$ for an angle of $4\pi+2T$, until the point $Z+re^{iT}$
  • straight line to $Re^{iT}$
  • run along $C_\infty$ for an angle of $-4\pi-2T$, until the point $Re^{-iT}$

Remark: The path $\gamma_0$ is easy to visualise.
After deforming it a tiny bit, it becomes isotopic to the image of the curve ⌘ under the map $z\mapsto (z-a)^{-1}$, where $a$ is the middle point of one of the lobes.

I believe that there exists an elliptic curve $E=\mathbb C/\Lambda$ and an immersion $$f:E\setminus\{0\}\to\mathbb C$$ such that $\forall T\in [0,\infty)$, the path $\gamma_T$ is the image under $f$ of a small counterclockwise loop (not quite round, but of bounded distortion) of radius $\simeq T^{-1/3}$ around the puncture.