A new approach to some nonlinear geometric equations in dimension two (Q2492662)

The authors consider several Liouville type results related to the equation \(-\Delta u = Ke^{2u}\) using an approach based on the holomorphic function associated with any solution. Namely, a solution gives rise to a holomorphic function which is zero if and only if the solution is canonical. It is reminiscent of the Hopf differential for harmonic maps from a surface. They apply this method to constant positive and nonpositive curvature surfaces with a boundary of constant geodesic curvature.