Topological regularity for solutions to the generalised Hopf equation

Abstract

The generalised Hopf equation is the first order nonlinear equation with data $\Phi$ a holomorphic functions and $\eta \ge 1$ a positive weight, \[ h_w\,\overline{h_\wbar}\,η(w) = Φ.\] 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.