Let $\Sigma^k$ be a $k$-dimensional Stein manifold with embedding as a real manifold (let's assume that that embedding is analytic if it makes things easier) $\Sigma^k \hookrightarrow \Bbb R^{2k}$.


*Main question.* Is it true that $\Sigma^2$ holomorphically embeds in $\Bbb C^2$?


*Addendum.* Is there an example of $\Sigma^k$ without embedding into $\Bbb C^k$ with $k < 7$? ($k > 1$ by Koebe uniformization theorem; on the other side, complex $7$-sphere ($\Bbb R^{14} \setminus \Bbb R^6$) does not embed in $\Bbb C^7$)



Stein manifolds with "wrong" minimal dimension of embedding
A result of [Gompf](https://arxiv.org/abs/1110.1865) shows that, in the case $k=2$, if the complex structure on the domain induced by a smooth embedding in $C^2$ is homotopic to a Stein structure, then the embedding is isotopic to a complex embedding. "Homotopic" means homotopic through almost-complex structures.


