Hilbert 16th problem via hyperbolic geometry

Question

More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a possible strategy is that a quadratic vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y)\end{cases}$$ can be rewritten in the form $z'=f(z,\bar{z}) $ with substitution $x=\frac{z+\bar{z}}{2}$ and $y=\frac{z-\bar{z}}{2i}$. He said that this possible strategy does not work for $n>2$. However, he thinks that this strategy leads to finitness of $H(2)$.

My immediate reaction at that time was the following: since hyperbolic geometry concerns the upper half plane, are we implicitly assuming that the upper half plane is invariant under flow? So are we assuming that we have an invariant line? If this is the case then the following fact is an obstruction for continuation:

Fact: every quadratic vector field with an invariant line has at most one limit cycle.

But I think that the story is more complicated. I guess that he was not assuming that the upper half plane is flow invariant. So I guess that there are some thing non trivial in this possible strategy.

I did not understand at all what is his strategy.I frequently asked him for more explanation. But I did not get any answer.

He allowed me to talk with others about his idea.

How does hyperbolic geometry can involve the Hilbert 16th problem? and how does this involvement work only for $n=2$ but not $n>2$.

Answer 1

1. This story is somewhat reminiscent of a (probably apocryphal) story of somebody writing a telegram to a mathematical research institute (Steklov Institute in Moscow, if I remember it correctly): "Solved Fermat's Last Theorem. Key idea - move $z^n$ to the left hand side. Details later."

2. No, hyperbolic geometry does not "concern the upper half plane". The upper half plane is just one of many models of the hyperbolic plane and the latter can appear in many different forms. (Besides, maybe hyperbolics space that the speaker had in mind is even the hyperbolic 3-space, or a Kobayashi-hyperbolic manifold, or Anosov-type dynamics, or whatever...)

3. There are some parallels of Hilbert-XVI and the Ahlfors Finiteness Theorem (and its generalizations) in the theory of Kleinian subgroups of $PSL(2,{\mathbb C})$ (so, maybe it is even the hyperbolic 3-space after all!). The parallels between the latter and holomorphic dynamics where exploited, for instance, by Dennis Sullivan - was he the speaker? - (dynamics of rational functions of one variable: proof of Fatou's Wandering Domain conjecture) and Xavier Gomez-Mont (a finiteness theorem for codimension 1 holomorphic foliations; see this 1980 JDG paper). The relation to hyperbolic geometry is only tangential; the idea is to prove a finiteness statement by arguing that otherwise, for some analytical reasons, a certain deformation space (say, a certain cohomology group) would have to be infinite-dimensional, while for some algebraic reasons such a space has to be finite-dimensional. For instance, in Sullivan's proof, the space of rational functions of the given degree is clearly finite-dimensional, while the existence of a wandering domain would create an infinite-dimensional space of quasiconformal deformations of such a function. The common feature of such proofs is that a certain PDE problem in one complex variable (maybe $z, \bar{z}$, to be more precise; for instance, the Beltrami equation) is well-posed, and these proofs break down in higher dimensions since the "right" PDE system turns out to be overdetermined.

4. The rest are details for somebody to figure out and collect his/hers Fields medal.