Analytic characterization of monotone Hopf-harmonics

TL;DR

This paper characterizes monotone Hopf-harmonics as solutions to a variational problem with boundary data, showing they are monotone if and only if their Jacobian determinant maintains a consistent sign, offering an alternative to harmonic homeomorphisms.

Contribution

It provides a complete characterization of monotone solutions to the inner-variational equation, linking monotonicity to the Jacobian determinant's sign, and explores their topological behavior via Hopf quadratic differentials.

Findings

Monotone Hopf-harmonics are solutions with non-changing Jacobian sign.

Monotonicity is equivalent to Jacobian sign preservation.

Topological analysis via Hopf quadratic differentials is crucial.

Abstract

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.