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The canonical lamination calibrated by a cohomology class

Aidan Backus

TL;DR

The paper constructs a canonical lamination $\lambda_\rho$ calibrated by every calibration in a unit-norm cohomology class $\rho\in H^{d-1}(M,\mathbb{R})$ on a closed oriented manifold $M$ with $2\le d\le 7$, showing its leaves are minimal hypersurfaces calibrated in $\rho$. It establishes a deep duality between calibrations and laminations, tying the geometry of $\lambda_\rho$ to the unit ball of the stable norm and introducing an earthquake-norm analogue in this infinitesimal setting. By developing the BV/least-gradient framework and ergodic theory of transverse measures, the authors prove that the dual flat $\rho^*$ is a polytope whose rational vertices correspond to closed leaves, and they provide bounds on irrational vertices via the first Betti number; they also derive strict convexity criteria linked to the fundamental group’s derived series, implying abundant uniquely ergodic calibrated laminations under topological constraints. The work draws a conceptual bridge to Teichmüller theory, offering a calculus of calibrations and laminations that constrains stable-norm geometry and hints at higher-dimensional extensions and singularity phenomena beyond $d=7$.

Abstract

Let $M$ be a closed oriented Riemannian manifold of dimension $2 \leq d \leq 7$, and let $ρ\in H^{d - 1}(M, \mathbb R)$ have unit norm. We construct a lamination $λ_ρ$ whose leaves are exactly the minimal hypersurfaces which are calibrated by every calibration in $ρ$. The geometry of $λ_ρ$ is closely related to the the geometry of the unit ball of the stable norm on $H_{d - 1}(M, \mathbb R)$, and so we deduce several results constraining the geometry of the stable norm ball in terms of the topology of $M$. These results establish a close analogy between the stable norm on $H_{d - 1}(M, \mathbb R)$ and the earthquake norm on the tangent space to Teichmüller space.

The canonical lamination calibrated by a cohomology class

TL;DR

The paper constructs a canonical lamination calibrated by every calibration in a unit-norm cohomology class on a closed oriented manifold with , showing its leaves are minimal hypersurfaces calibrated in . It establishes a deep duality between calibrations and laminations, tying the geometry of to the unit ball of the stable norm and introducing an earthquake-norm analogue in this infinitesimal setting. By developing the BV/least-gradient framework and ergodic theory of transverse measures, the authors prove that the dual flat is a polytope whose rational vertices correspond to closed leaves, and they provide bounds on irrational vertices via the first Betti number; they also derive strict convexity criteria linked to the fundamental group’s derived series, implying abundant uniquely ergodic calibrated laminations under topological constraints. The work draws a conceptual bridge to Teichmüller theory, offering a calculus of calibrations and laminations that constrains stable-norm geometry and hints at higher-dimensional extensions and singularity phenomena beyond .

Abstract

Let be a closed oriented Riemannian manifold of dimension , and let have unit norm. We construct a lamination whose leaves are exactly the minimal hypersurfaces which are calibrated by every calibration in . The geometry of is closely related to the the geometry of the unit ball of the stable norm on , and so we deduce several results constraining the geometry of the stable norm ball in terms of the topology of . These results establish a close analogy between the stable norm on and the earthquake norm on the tangent space to Teichmüller space.

Paper Structure

This paper contains 20 sections, 38 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Differential forms in L-infinity
  5. Calibrated geometry
  6. Laminations
  7. Perimeter-minimizing sets
  8. Calibrated laminations and functions of least gradient
  9. Calibrated laminations
  10. Functions of least gradient
  11. Duality of calibrations and laminations
  12. Construction of the canonical lamination
  13. Transverse measures on the canonical lamination
  14. Ergodic theory of the canonical lamination
  15. The Arnoux--Levitt lemma
  16. ...and 5 more sections

Key Result

Theorem 1.1

For every $g \geq 2$ and $\rho, \sigma \in \mathscr T_g$, there exists a unique largest chain-recurrent geodesic lamination $\lambda_{\rho, \sigma}$ in $(\Sigma_g, \rho)$ such that for every Lipschitz map $f: (\Sigma_g, \rho) \to (\Sigma_g, \sigma)$ homotopic to $\mathop{\mathrm{id}}\nolimits_{\Sigm

Theorems & Definitions (84)

  • Theorem 1.1: Gu_ritaud_2017
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7: daskalopoulos2025
  • Theorem 2.1: $L^\infty$ Poincaré lemma
  • proof
  • Theorem 2.2: normal trace theorem
  • ...and 74 more