As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a holomorphic immersion into the complex plane. Another example is given by the theorem of Behnke and Stein, which says that every connected open Riemann surface is a Stein manifold. Is there any more interesting results for open Riemann surfaces?
[Math] Interesting results for open Riemann surfaces
complex-geometrycv.complex-variables
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I think Hunter and Greg's answers make it hard to see the forest for the trees. Let X be a compact Riem. surface of genus >= g. Let Y be the universal cover of X equipped with the complex structure pulled back from X. As a complex manifold, Y is isomorphic to the upper half plane, and the deck transformations form a subgroup Gamma of PSL_2(R). There will be characters chi of Gamma for which there are nonzero functions f on Y such that f(gz) = chi(g) f(z). For chi ample enough (not defined here), we will be able to choose functions (f_1, f_2, f_3) such that z --> (f_1(z) : f_2(z) : f_3(z)) gives an immersion X --> P^2. All of this works in any genus.
The technical issue is that this map is an immersion, not an injection, meaning that the image can pass through itself. One can either decide to live with this, or work with maps to P^3 instead.
Most books that I have seen don't lift all the way to the universal cover of X. Instead, they take the covering of X which corresponds to the commutator subgroup of pi_1(X). This can be motivated in a particular nice way in terms of the Jacobian. This is a complex manifold with the topological structure of a 2g dimensional torus. There is a map X --> J, so that the map pi_1(X) --> pi_1(J) is precisely the map from pi_1(X) to its abelianization. People then work with the universal cover of J, and the preimage of X inside it. This has three advantages: the universal cover of J is C^g, not the upper half plane; the group of Deck transformations is Z^{2g}, not the fundamental group of a surface, and the action on C^g is by traslations, not Mobius transformations. The functions which transform by characters, in this setting, are called Theta functions*, and they are given by explicit Fourier series.
*This is a slight lie. Theta functions come from a certain central extension of the group of Deck transformations. It is certain ratios of Theta functions that will transform by characters as sketched above. The P function itself, for example, is a ratio of four Theta functions. In the higher genus case, in my limited reading, I haven't seen names for these ratios, only for the Theta functions.
The answer is negative. Suppose for contradiction that $S$ is such a surface, and let me first assume that it is smooth and projective.
Fix $g\geq 24$. Then the coarse moduli space of genus $g$ curves $M_g$ is of general type (this is due to Harris, Mumford and Eisenbud, see for instance [The Kodaira dimension of the moduli space of curves of genus $\geq 23$]), hence a fortiori not uniruled. Let $H$ be the Hilbert scheme of smooth curves of genus $g$ in $S$ and let $(H_i)_{i\in I}$ be its irreducible components: the index set $I$ is countable. By hypothesis, the classifying morphism $H\to M_g$ is surjective at the level of $\mathbb{C}$-points. By a Baire category argument, there exists $i\in I$ such that $H_i\to M_g$ is dominant.
Suppose that the natural morphism $H_i\to \operatorname{Pic}(S)$ is constant. Then $H_i$ is an open subset of a linear system on $S$, hence is covered by (open subsets of) rational curves. By our choice of $g$, $H_i\to M_g$ cannot be dominant, which is a contradiction. Thus, $H_i\to \operatorname{Pic}(S)$ cannot be constant, which proves that $\operatorname{Pic}^0(S)$ cannot be trivial. Equivalently, the Albanese variety $A$ of $S$ is not trivial.
Consider the Albanese morphism $a:S\to A$. The curves embedded in $S$ are either contracted by $a$ or have a non-trivial morphism to $A$. Those that are contracted by $a$ form a bounded family, hence have bounded genus. Curves $C$ that have a non-trivial morphism to $A$ are such that there is a non-trivial morphism $\operatorname{Jac}(C)\to A$, but this is impossible if $\operatorname{Jac}(C)$ is simple of dimension $>\dim(A)$. Consequently, a smooth curve with simple jacobian that has high enough genus cannot be embedded in $S$. This concludes because a very general curve of genus $g$ has simple jacobian (there even exist hyperelliptic such curves by [Zarhin, Hyperelliptic jacobians without complex multiplication]).
As pointed out in the comments, the argument needs to be modified if $S$ is non-algebraic or singular. I explain now these additional arguments.
If $S$ is smooth but non-algebraic, we can use the following (probably overkill) variant. We can consider the open space $H$ of the Douady space of $S$ parametrizing smooth connected curves in $S$. It is a countable union of quasiprojective varieties by [Fujiki, Countability of the Douady space of a complex space] and [Fujiki, Projectivity of the space of divisors on a normal compact complex space]. It has an analytic morphism to the analytic space $\operatorname{Pic}(S)$ parametrizing line bundles on $S$ [Grothendieck, Techniques de construction en géométrie analytique IX §3]. The dimension of $\operatorname{Pic}(S)$ is finite equal to $h^1(S,\mathcal{O}_S)$. The dimension of the fibers of $H\to\operatorname{Pic}(S)$ is at most $1$. Indeed, otherwise, we would have a linear system of dimension $>1$ on $S$ consisting generically of smooth connected curves, hence a dominant rational map $S\dashrightarrow\mathbb{P}^2$, showing that $S$ is of algebraic dimension $2$, hence that $S$ is algebraic. It follows that every connected component of $H$ has dimension $\leq h^1(S,\mathcal{O}_S)+1$. Now choose $g$ such that $\dim(M_g)>h^1(S,\mathcal{O}_S)+1$, and let $H_g$ be the union of connected components of $H$ parametrizing genus $g$ curves. A Baire category argument applied to the image of $H_g\to M_g$ shows that there are genus $g$ curves that do not embed in $S$, as wanted.
Finally, if $S$ is singular, consider a desingularization $\tilde{S}\to S$. The hypothesis on S implies that every smooth projective curve may be embedded in $\tilde{S}$, with the exception of at most a finite number of isomorphism classes of curves (namely, the curves that are connected components of the locus over which $\tilde{S}\to S$ is not an isomorphism). The arguments above apply as well in this situation.
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The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911, MR0228671, MR0159935, MR0264064, MR1973182. It is true that there are "too many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838, and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581.