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 Ω ⊂ C , with data Φ a holomorphic function and η≥ 1 a positive weight on Ω , hwhw¯¯η(w)=Φ. The Hopf equation is the special case η(w) = η~ (h(w)) and reflects that h is harmonic with respect to the conformal metric η~(z)|dz| , usually η 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.