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Minimal diffeomorphisms with $L^1$ Hopf differentials

Nathaniel Sagman

Abstract

We prove that for any two Riemannian metrics $σ_1, σ_2$ on the unit disk, a homeomorphism $\partial\mathbb{D}\to\partial\mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(\mathbb{D},σ_1)\to (\mathbb{D},σ_2)$ with $L^1$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^1$ assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees.

Minimal diffeomorphisms with $L^1$ Hopf differentials

Abstract

We prove that for any two Riemannian metrics on the unit disk, a homeomorphism extends to at most one quasiconformal minimal diffeomorphism with Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees.
Paper Structure (20 sections, 16 theorems, 44 equations, 2 figures)

This paper contains 20 sections, 16 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Minimal diffeomorphisms
  3. The new main inequality
  4. Outline of the argument
  5. Theorem A
  6. Theorem B
  7. Future perspectives
  8. Acknowledgements
  9. Harmonic maps
  10. Harmonic maps
  11. Harmonic maps to $\mathbb R$-trees
  12. Proofs
  13. The Reich-Strebel formula
  14. The Plateau problem in a product of trees
  15. The Plateau problem from a polynomial
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\varphi:\partial\mathbb{D}\to\partial \mathbb{D}$ be a quasisymmetric map and let $\sigma$ be a hyperbolic metric on $\mathbb{D}$. There exists a unique minimal diffeomorphism $f:(\mathbb{D},\sigma)\to (\mathbb{D},\sigma)$ that extends to $\varphi$ on $\partial\mathbb{D}.$ Moreover, $f$ is quas

Figures (2)

  • Figure 1: The foliation and $\mathbb R$-tree of a polynomial with two double order zeros, restricted to $\overline{\mathbb{D}}$.
  • Figure 2: An embedded circle bounding a disk in a particularly simple product of simplicial complexes.

Theorems & Definitions (46)

  • Theorem 1.1: Bonsante-Schlenker
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Proposition 1.5
  • Definition 2.1
  • Theorem 2.2: Theorem 2.2 in KS
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 36 more