Topology and Complex Structures of Leaves of Foliations by Riemann Surfaces

Abstract

We study conformal structure and topology of leaves of singular foliations by Riemann surfaces.

Ergodic theorems for laminations and foliations: recent results and perspectives

“…Consequently, holomorphic foliations in P k are generically Brody hyperbolic, see Theorem 2.34 (1). The reader may find in [89] a nice discussion on the topology and the conformal structures of leaves of a singular holomorphic foliation which is Brody hyperbolic.…”

Some Open Problems on Holomorphic Foliation Theory

“…The problem was discussed by Ilyashenko [19]. There are partial results for B d by Sibony-Wold in [28]. For the case of A d , the question was discussed by Shcherbakov, Rosales-González, Ortiz-Bobadilla in [26].…”

Injective Hyperbolicity for quotients of balls and polydisks

“…In complex dimension one, i.e., in the the case that X is a Riemann surface, the corresponding problem is quite simple. If X is hyperbolic the metrics coincide if and only if X is the unit disk (see [10] where the injective Kobayashi metric was introduced on foliations), if X = C or X = P 1 both metrics vanish identically, and if X = C * or X is a torus, the metrics are different due to the Koebe 1 4 -theorem. Furthermore, in complex dimension larger than 2, the metrics always coincide, see [9].…”