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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

##### MSC:
 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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