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Question: Can every non-compact Riemann surface be holomorphically embedded into $\mathbb{C}^2$? If not, what are some (all?) of the obstructions to such an embedding?

This question is partially inspired by the Wikipedia page on Stein manifolds, which taught me two things:

  • Behnke-Stein Theorem (1948): Every non-compact Riemann surface is Stein, hence can be holomorphically embedded in some $\mathbb{C}^N$.

  • Every Stein manifold of complex dimension $n$ can be embedded into $\mathbb{C}^{2n+1}$ by a biholomorphic proper map. (It would be nice to have a citation for this.)

Together, these two theorems imply that every non-compact Riemann surface holomorphically embeds into $ \mathbb{C}^3$. This raises the question of embedding into $\mathbb{C}^2$.

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    $\begingroup$ A citation for the second theorem: E. Bishop, Mappings of partially analytic spaces, Amer. J. Math. 83, 209-242 (1961); and R. Narasimhan, Imbedding of holomorphically complete convex spaces, Amer. J. Math. 82, 917-934 (1960). These appear to be the foundational papers on this (can't dig them out of the library right now, sorry!), found via Cieliebak and Eliashberg's book From Stein to Weinstein and Back, specifically Chapter 5, theorem 15. I would recommend paging through the book, too - it can be found on Cieliebak's website. $\endgroup$ – KReiser Feb 2 at 6:52
  • $\begingroup$ Related question: mathoverflow.net/questions/221957/… $\endgroup$ – Prism Feb 9 at 15:43
  • $\begingroup$ I found this Stein manifolds.pdf to show an immersion into $\mathbb{C}^N$ exists they simply say that for any compact $K \subset X $ and $N \ge 2\dim(X)+1$ the subset of $\mathcal{O}(X,\mathbb{C}^N)$ that are not injective $K \to \mathbb{C}^N$ is closed with empty interior, thus the subset of $\mathcal{O}(X,\mathbb{C}^N)$ that are not injective $X \to \mathbb{C}^N$ is contained in a countably infinite union of closed sets with empty interior, so it is not the whole of $\mathcal{O}(X,\mathbb{C}^N)$. $\endgroup$ – reuns Feb 9 at 16:45
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This is an important open problem going back to Forster, Bell and Narasimhan. To quote from

A. Alarcón, F. Forstnerič, Every bordered Riemann surface is a complete proper curve in a ball, Mathematische Annalen, Vol. 357 (2013) Issue 3, 1049–1070.

"It is classical that any open Riemann surface immerses properly holomorphically into ${\mathbb C}^2$, but it is an open question whether it embeds into ${\mathbb C}^2$."

Existence of proper and non-proper embeddings are both unknown. Many things are known. For instance, the complement to any finite collection of pairwise disjoint closed topological disks in a genus 1 compact Riemann surface (aka a torus or an elliptic curve) embeds:

E. F. Wold, Embedding subsets of tori properly into ${\mathbb C}^2$. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1537–1555.

It is also known that there are no topological obstructions to a holomorphic embedding of an open Riemann surface in ${\mathbb C}^2$:

A. Alarcón, F. López, Proper holomorphic embeddings of Riemann surfaces with arbitrary topology into ${\mathbb C}^2$. J. Geom. Anal. 23 (2013), no. 4, 1794–1805.

and

A. Alarcón, J. Globevnik, Complete embedded complex curves in the ball of ${\mathbb C}^2$ can have any topology. Anal. PDE 10 (2017), no. 8, 1987–1999.

See these slides for further background on this problem.

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