Analytic characterization of monotone Hopf-harmonics

We study solutions of the inner-variational equation associated with the Dirichlet energy in the plane, given homeomorphic Sobolev boundary data. We prove that such a solution is monotone if and only if its Jacobian determinant does not change sign. These solutions, called monotone Hopf-harmonics, are a natural alternative to harmonic homeomorphisms. Examining the topological behavior of a solution (not a priori monotone) on the trajectories of Hopf quadratic differentials plays a sizable role in our arguments.