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

Camacho, C.; Lins Neto, Alcides; Sad, P.

Publ. Math., Inst. Hautes Étud. Sci. 68, 187-203 (1988).

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)

Keywords:

holomorphic foliation; foliation with singularities of \({\mathbb{C}}P(2)\); polynomial differential equation; nontrivial minimal set; leaves; exponential growth; hyperbolic Riemannian surfaces; parabolic ends

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