Minimal Diffeomorphisms with L1 Hopf Differentials

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.