Topological regularity for solutions to the generalised Hopf equation

The generalised Hopf equation is the first order nonlinear equation with data $\mathrm{\Phi }$ a holomorphic functions and $\eta \ge 1$ a positive weight, $$h_w\overline{h_{\text{wbar}}}\eta (w)=\mathrm{\Phi }\mathrm{.}$$ 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 $\sqrt{\tilde{\eta }(z)}\mathrm{\mid }dz\mathrm{\mid }$. This article obtains conditions on the data to ensure that a solution is open and discrete. We also prove a strong uniqueness result.