Uniformization of the leaves of a rational vector field

Structure of leaves and the complex Kupka-Smale property

“…It is known that any leaf of a generic polynomial foliation of degree n is hyperbolic [3], [8], [16]. We expect that the same answer is true for generic foliations of Stein manifolds and that technique from [3], [16], [8] can be adjusted to attack the problem.…”

“…We expect that the same answer is true for generic foliations of Stein manifolds and that technique from [3], [16], [8] can be adjusted to attack the problem. See the paper [13] for a vast discussion of open problems.…”

Oseledec multiplicative ergodic theorem for laminations

“…In this context, the holonomy of leaves is closely related to the uniformizations of leaves and their Poincaré metric. This subject has received a lot of attention in the recent years (see, for example, the works by Candel [2], Candel-Gómez Mont [6], Dinh-Nguyen-Sibony [12,13,14], Fornaess-Sibony [16,17,18], Neto [28] etc). We also hope that the results of this Memoir may find applications in the dynamics of moduli spaces and in the geometric dynamics of laminations and foliations.…”

Topology and Complex Structures of Leaves of Foliations by Riemann Surfaces

“…Proof. For such a foliations all leaves are hyperbolic, and the Kobayashi metric is continuous (see [3] and the survey [11]). Near any point p ∈ E, all leaves except finitely many separatrices are simply connected.…”