In a current project with a colleague, we have come across the following reasonably classical-sounding geometric question. While not vital to our work, it would be interesting if anyone has seen this type of issue discussed before, and if there are any references.

To keep things simple, I will formulate a special case, which appears to retain all issues present in the general case.

## Question

Let $\gamma:[0,\infty)\to\mathbb{C}\setminus\{0\}$ be a continuous and injective curve in the punctured complex number plane, with $\gamma(t)\to\infty$ as $t\to\infty$.

Is there a simply-connected domain $V\subset\mathbb{C}\setminus\{0\}$ with $\gamma\subset V$ such that $\gamma$ tends to the boundary of $V$ *horocyclically* in $V$?

(The latter condition means the following: There is a conformal isomorphism $\phi$ that maps $V$ to the right half plane in such a way that $\operatorname{Re}(\phi(\gamma(t)))\to+\infty$ as $t\to\infty$.)

## Remarks

- We may additionally suppose that a function $\delta(t)$ is given, and require that $V\subset\bigcup_{t\geq 0}B(\gamma(t),\delta(t))$, where $B(z,\delta)$ is the open disk of radius $\delta$ around $z$. This makes the statement of the question slightly more complicated, but makes the discussion of examples less cumbersome. (It is also part of the more general setup I mentioned above, which additionally replaces $\mathbb{C}\setminus\{0\}$ by an arbitrary open Riemann surface.)
- If the curve $\gamma$ is $C^1$, then it is easy to construct such a domain, by taking $V$ to be a (shrinking) "tubular" neighborhood of $\gamma$. In this case, it is even possible to ensure that the convergence is
*non-tangential*, meaning that the argument of $\phi(\gamma(t))$ stays bounded as $t\to\infty$. - It is not too difficult to see that non-tangential convergence cannot be obtained without some regularity assumption. However, perhaps a suitable uniform Lipschitz condition is sufficient.
- However, for horocyclic convergence, one can easily make do with much weaker conditions. For example, it seems enough to ask that the curve is $C^1$ on some sequence of intervals tending to infinity, while the behaviour in between can be as bad as you like.
- I tend to think that one can probably construct a counterexample for horocyclic convergence, but I may be wrong and in any case it is likely to involve some thought and effort. So I am hoping that someone can tell me that this or a similar question has been discussed in the literature.