Topological Regularity for Solutions to the Generalised Hopf Equation

Abstract

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