Topological regularity for solutions to the generalised Hopf equation

The generalised Hopf equation is the first order nonlinear equation with data $\Phi$ a holomorphic functions and $\eta\geq 1$ a positive weight, $$h_w\,\overline{h_\wbar}\,\eta(w) = \Phi.$$ The Hopf equation is the special case $\eta(w)=\tilde{\eta}(h(w))$ and reflects that $h$ is harmonic with respect to the conformal metric $\sqrt{\tilde{\eta}(z)}|dz|$. This article obtains conditions on the data to ensure that a solution is open and discrete. We also prove a strong uniqueness result.