Minimal diffeomorphisms with $L^1$ Hopf differentials

We prove that for any two Riemannian metrics ${\sigma }_1,{\sigma }_2$ on the unit disk, a homeomorphism $\mathrm{\partial }\mathbb{D}\to \mathrm{\partial }\mathbb{D}$ 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.