Leafwise smoothing laminations

Leafwise Homotopy Equivalences and Leafwise Sobolov Spaces

We prove that a leafwise homotopy equivalence between compact foliated manifolds induces a well defined bounded operator between all Sobolov spaces of leafwise (for the natural foliations of the graphs of the original foliations) differential forms with coefficients in a leafwise flat bundle. We further prove that the associated map on the leafwise reduced L2 cohomology is an isomorphism which ...

Leafwise Holomorphic Functions

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

R-trees and laminations for free groups I: Algebraic laminations

This paper is the first of a sequence of three papers, where the concept of an Rtree dual to (the lift to the universal covering of) a measured geodesic lamination L in a hyperbolic surface S is generalized to arbitrary R-trees provided with a (very small) action of the free group FN of finite rank N ≥ 2 by isometries. In [CHL-II] to any such R-tree T a dual algebraic lamination L(T ) is associ...

Non-euclidean Affine Laminations

The purpose of the present paper is to discuss examples of aane Riemann surface laminations which do not admit a leafwise Euclidean structure. The rst example of such a lamination was constructed by Ghys Gh97]. Our discussion is based on the geometric methods developed by Lyubich, Minsky and the author LM97], KL01], which rely on the observation that any aane surface A gives rise in a natural w...

Laminations in Holomorphic Dynamics