Compressible ends of leaves in foliated 3-manifolds

Abstract

In this paper we study the asymptotic behavior of cylindrical ends in compact foliated 3-manifolds and give a sufficient condition for these ends to spiral onto a toral leaf.

A Generalization of the Poincaré--Bendixson Theorem for Compact 3-manifolds

“…But the existence of such an arc α violates the transverse orientability of F. Hence, Proposition 6 is valid and our theorem follows easily. In [3] the existence of such a sphere S 0 and of such an arc α is immediate from the hypothesis that the end is of trivial linking type. In the present work, as no assumption on the cylindrical compressible ends is imposed, we prove the existence of S 0 and of α and this proof constitutes the real core of this paper.…”

“…The aim of the present work is to generalize the main theorem of [3] by removing assumptions essential to its proof there. Moreover, the main assumptions of [1,2,7], namely that the leaves are either totally proper or of nonexponential growth, are replaced by a purely topological condition which makes the type of ends completely analogous to the type of orbits in dimension 2.…”

“…For the proof of our theorem we use certain partial results from [3], stated here without proofs, as well as new ones. Note that in order to prove the main result in [3], we have assumed that the cylindrical compressible end is of trivial linking type (see Definition 3 in [3]).…”

“…Note that in order to prove the main result in [3], we have assumed that the cylindrical compressible end is of trivial linking type (see Definition 3 in [3]). In fact, this hypothesis is very strong and nonnatural in the sense that is practically impossible to be verified.…”

“…In the second section we prove the main theorem. Roughly speaking, the proof of the theorem, as well as the proof of the theorem in [3], is based on the following argument: Assuming that Proposition 6 below is not valid we construct a sphere S 0 embedded in M and an arc α which is transverse to F and which has its endpoints on S 0 . But the existence of such an arc α violates the transverse orientability of F. Hence, Proposition 6 is valid and our theorem follows easily.…”