Minimal Diffeomorphisms with L1 Hopf Differentials

We prove that for any two Riemannian metrics σ ₁, σ ₂ on the unit disk, a homeomorphism ∂ D→ ∂ D extends to at most one quasiconformal minimal diffeomorphism (D,σ ₁)→ (D,σ ₂) with L¹ 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 L¹ 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.