Let $C$ be a smooth analytic curve (holomorphic submanifold of complex dimension one) in the complex plane $\mathbb{C}^2$. What geometric or topological criteria guarantee that $C$ is algebraic? For example, is there some condition on the second fundamental form that guarantees algebraicity?

$\begingroup$ By "curve" you mean a real analytic curve, i.e. submanifold of real dimension 1? By "complex plane" do you mean $\mathbb{C}$ or $\mathbb{C}^2$? $\endgroup$– Ben McKayMay 6 '17 at 18:08

$\begingroup$ Edited for clarity as suggested. $\endgroup$– Chris WoodwardMay 6 '17 at 18:12

4$\begingroup$ @ChrisWoodward: You haven't made any completeness assumptions, so a priori we could be talking about a small piece of a holomorphic curve that may or may not be algebraic. The specific data that you know about the curve (for example, you might only know the induced metric or only the second fundamental form) may or may not be enough to determine whether it is algebraic. $\endgroup$– Robert BryantMay 7 '17 at 9:26

5$\begingroup$ @ChrisWoodward: If you know that it is complete, connected, and that the integral of the Gauss curvature is finite, then it is algebraic. I think this is originally due to Osserman. Since the Gauss curvature is essentially the (negative of) the norm of the second fundamental form, I guess that sufficiently fast decay of the second fundamental form would be enough to guarantee that the integral of the Gauss curvature is finite. $\endgroup$– Robert BryantMay 7 '17 at 17:10

3$\begingroup$ @alvarezpaiva: Yes, since 'algebraic' is invariant under projective transformations, and 'completeness' doesn't really matter as long as you have connectedness. For any local holomorphic curve in $\mathbb{CP}^2$ that is not linear, there defined is a projectively invariant meromorphic cubic form, which vanishes if and only if the curve is (part of) a conic. For curves that are not (part of) a line or a conic, there also is a projectively invariant meromorphic octic form, and the curves of algebraic degree $d$ are characterized by a certain differential equation $D_d$ relating the two. $\endgroup$– Robert BryantMay 9 '17 at 18:47
I do not have an answer to this question, but it reminds me of a similar question for minimal surfaces in the euclidean 3space (they are locally the real part of a null holomorphic map into $\mathbb R^3$). In that case, if you have an embedded complete end of finite total curvature, the Riemann surface structure (around the end) is that of a punctured disc. Therefore, complete minimal surfaces of finite total curvature with embedded ends leads to a algebraic Weierstrass data (the null curve (with imaginary periods) is given by two meromorphic spinor fields). I guess that some of that theory can be translated to the case of holomorphic curves in $\mathbb C^2.$ A place to read about that is the book of Osserman on Minimal surfaces.

$\begingroup$ Thanks, that is the kind of result I was looking for, although as you say in $\mathbb{C}^2$ rather than $\mathbb{R}^3$. I will check it out. $\endgroup$ May 7 '17 at 12:48