Minimal sets of foliations on complex projective spaces. (English) Zbl 0682.57012

Let \({\mathcal F}\) be a foliation (with a finite number of singularities) of \({\mathbb{C}}P(2)\) defined by a polynomial differential equation \(P(x,y)dy- Q(x,y)dx=0\) on \({\mathbb{C}}^ 2\) with P, Q relatively prime. The authors prove that \({\mathcal F}\) has at most one nontrivial minimal set, and under some generic condition imposed on the singularities, all the leaves accumulate on that set. It is shown that the leaves of a nontrivial minimal set have exponential growth, they are hyperbolic Riemannian surfaces in a suitable Hermitian metric in \({\mathbb{C}}P(2)\setminus \sin g({\mathcal F})\), and they have no parabolic ends.
Reviewer: A.Piatkowski


57R30 Foliations in differential topology; geometric theory
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
53C12 Foliations (differential geometric aspects)
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