From pre-lamination to foliated plane

TL;DR

This paper characterizes which pre-laminations on the circle can serve as boundary at infinity for foliations on the plane and constructs the unique foliation corresponding to each such pre-lamination.

Contribution

It provides a complete solution to identifying and constructing foliations from given boundary pre-laminations on the circle, including singular cases.

Findings

Characterization of boundary pre-laminations for plane foliations

Construction method for foliations from boundary pre-laminations

Extension to singular foliations with prong singularities

Abstract

To a singular foliation on the plane corresponds a circular boundary at infinity endowed with a pre-lamination on the circle. We solve the converse direction. We determine which pre-lamination on the circle are boundary at infinity of a foliation, and we build the corresponding (unique) foliation. We consider both regular foliations and a singular foliations with prong singularities.