Topological Regularity for Solutions to the Generalised Hopf Equation

Abstract

The generalised Hopf equation is the first order nonlinear equation defined on a planar domain Omega subset {mathbb {C}}, with data Phi a holomorphic function and eta ge 1 a positive weight on Omega , hwhw¯¯η(w)=Φ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} h_w\\,\\overline{h_{\\overline{w}}}\\,\\eta (w) = \\Phi . \\end{aligned}$$\\end{document}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|, usually eta is 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.