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From pre-lamination to foliated plane

Christian Bonatti, Théo Marty

TL;DR

This work solves the inverse problem of end laminations: given a pre-lamination on the circle, determine when it is the boundary at infinity of a foliation of the plane and construct the corresponding foliation. It develops a unified framework based on planar structures, end pre-laminations, and the notions of shell/star regions to characterize realizable pre-laminations for both non-singular and singular (pA) foliations. A singular Kaplan-type theory is established, proving that every singular planar structure arises from a pA-foliation and is unique up to homeomorphism, with universal models built via ramified covers and surgeries. The results provide a principled path from circle data to concrete foliations, enabling a reconstruction paradigm with potential applications to group actions and dynamics on surfaces. Overall, the paper deepens the correspondence between end laminations, leaf-space topology, and foliation realization on the plane.

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.

From pre-lamination to foliated plane

TL;DR

This work solves the inverse problem of end laminations: given a pre-lamination on the circle, determine when it is the boundary at infinity of a foliation of the plane and construct the corresponding foliation. It develops a unified framework based on planar structures, end pre-laminations, and the notions of shell/star regions to characterize realizable pre-laminations for both non-singular and singular (pA) foliations. A singular Kaplan-type theory is established, proving that every singular planar structure arises from a pA-foliation and is unique up to homeomorphism, with universal models built via ramified covers and surgeries. The results provide a principled path from circle data to concrete foliations, enabling a reconstruction paradigm with potential applications to group actions and dynamics on surfaces. Overall, the paper deepens the correspondence between end laminations, leaf-space topology, and foliation realization on the plane.

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.

Paper Structure

This paper contains 50 sections, 75 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Short presentation of the questions and results
  3. Statement of our results: the non-singular case
  4. Statement of our results: the general (singular) case
  5. Choice of leaf space.
  6. Acknowledgment.
  7. Preliminary notions
  8. Ramified cover
  9. Singular foliations: terminology
  10. Non-separated leaves, regular leaves.
  11. The circle at infinity of a foliation
  12. Planar structure
  13. Tame non-Hausdorff $1$-manifold
  14. (Non-singular) planar structure
  15. Singular planar structure
  16. ...and 35 more sections

Key Result

Theorem 1

Let $L$ be a pre-lamination of $S^1$ whose endpoints are dense in $S^1$ and with no more than countably many leaves ending at a given endpoints. Then $L$ is the end pre-lamination of a (non singular) foliation of $\mathop{\mathrm{Int}}\nolimits(D^2)$ if and only if it is a shell pre-lamination and h

Figures (2)

  • Figure 1: Branchings from foliations to pre-laminations.
  • Figure 2: $L$-interval and transverse curve.

Theorems & Definitions (102)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5: MatherBonattiCirc
  • Remark 2.6
  • ...and 92 more