Leafwise Quasigeodesic Foliation in Dimension Three and the Funnel Property

TL;DR

This paper constructs specific 1-dimensional foliations within 3-manifolds that are quasigeodesic in each leaf but do not satisfy the funnel property, addressing a key question in foliation theory.

Contribution

It provides a counterexample showing that leafwise quasigeodesic property does not imply the funnel property in 3-manifold foliations.

Findings

Constructed foliations are quasigeodesic but not funnel in most leaves.

Demonstrated that funnel property is not a consequence of leafwise quasigeodesic property.

Answered an open question in foliation theory.

Abstract

We construct one dimensional foliations which are subfoliations of two dimensional foliations in 3-manifolds. The subfoliation is by quasigeodesics in each two dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves. This article answers the question whether the funnel property is a consequence of the leafwise quasigeodesic property or not.