Minimal diffeomorphisms with $L^1$ Hopf differentials

TL;DR

This paper establishes the uniqueness of minimal diffeomorphisms with $L^1$ Hopf differentials between Riemannian disks, extending boundary homeomorphisms, and introduces a novel proof avoiding anti-de Sitter geometry.

Contribution

It proves the uniqueness of quasiconformal minimal diffeomorphisms with $L^1$ Hopf differentials between Riemannian disks, without relying on anti-de Sitter geometry.

Findings

Uniqueness of minimal diffeomorphisms with $L^1$ Hopf differentials

Failure of the result without the $L^1$ assumption in variable curvature

Solution uniqueness for a Plateau problem in a product of trees

Abstract

We prove that for any two Riemannian metrics $\sigma_1, \sigma_2$ on the unit disk, a homeomorphism $\partial\mathbb{D}\to\partial\mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(\mathbb{D},\sigma_1)\to (\mathbb{D},\sigma_2)$ 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.