How many points of a sequence can we catch with an analytic disc?

Question

Let $X\subset \mathbb{C}^{n}$ be a domain. You can assume that it is nice (e.g. bounded convex balanced ). Let $\{x_n\}$ be a sequence of points that does not have a limit point in $X$.

Let $D$ be the (unit) disc on the plane.

Is there a holomorphic $\varphi:D\to X$ such that $\varphi(D)$ contains an infinite number of $x_n$?

Of course there are further questions: can we catch all the points? Or maybe all but finite number?

Answer 1

The answer to your quesiton is yes.

In particular in Discs in Stein manifolds containing given discrete sets B. Drinovec Drnovšek proves that given a discrete subset $S$ of a connected Stein manifold $M$ (every domain of holomorphy in $\mathbb{C}^m$ is if this kind) there is a proper holomorphic map $f:D \rightarrow M$ such that $S\subset f(D)$. Moreover if dim $M\geq$3 the map f can be chosen to be an embedding.