Topological Regularity for Solutions to the Generalised Hopf Equation

Abstract The generalised Hopf equation is the first order nonlinear defined on a planar domain $$\Omega \subset {\mathbb {C}}$$ Ω ⊂ C , with data $$\Phi $$ Φ holomorphic function and $$\eta \ge 1$$ η ≥ 1 positive weight $$\begin{aligned} h_w\,\overline{h_{\overline{w}}}\,\eta (w) = \Phi . \end{aligned}$$ h w ¯ ( ) = . special case (w)={\tilde{\eta }}(h(w))$$ ~ reflects that h harmonic respect to conformal metric $$\sqrt{{\tilde{\eta }}(z)}|dz|$$ z | d usually hyperbolic metric. This article obtains conditions ensure solution open discrete. We also prove strong uniqueness result.