# The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$

It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains a real limit cycle. This leaf is a Riemann surface. What can be said about the nature(type) of this leaf as a Riemann surface(Hyperbolic? Parabolic? elliptic?)

Well, in general not too much can be said without further information. For sure the leaf has a nontrivial fundamental group, so not an elliptic Reiamann surface (i.e. a Riemann sphere). But it could be a punctured Riemann sphere. Every polynomial vector field on $\mathbb{C}^2$ extends to a holomorphic foliation on $\mathbb{CP}^2$ with singularities. The complex line at infinity is a Riemann sphere and (with a number of points removed, which are the singularities of the foliation at infinity) is a leaf of a generic polynomial vector field. To get an idea about the topology of the leaves of the foliation, you can take a cross-section to the sphere at infinity (away from the singularities) and by lifting the elements of the fundamental group of the sphere, minus the singularities, on the leaves that go through the cross-section, you can get a representation of the group into the (pseud-)group of holomorphic diffeomorphisms on the cross-section (or if you prefer, the group of holomorphic germs of the cross-section). I think, generically, this group will have very complicated orbits (one orbit represents the topology of one leaf), and if I am not mistaken, most of the orbits are dense, which means generically, the leaves of the foliation are densely immersed in $\mathbb{CP}^2$. Some of them will contain closed sub-orbits, corresponding to cycles in the direction of one element of the fundamental group, but the rest of the orbit could be dense. So you may have dense cylinders, or even more complicated densely immersed non-trivial Riemann surfaces. So generically, hyperbolic Riemann surfaces.