Hilbert 16th problem via hyperbolic geometry 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  $n>2$.

 A: *

*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."  

*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...) 

*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. 

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