Liouville's formula in arbitrary planar domains
This paper deals with the Liouville equation \[ \Delta_x u(x,y)= Ke^{-2u(x,y)},\quad (x,y)\in \Omega,\tag{1} \] where \(K\) is a real constant, \(u\) is a real \(C^2\) function and \(\Omega\subset\mathbb R^2\) is a plane domain. The aim of this work is to extend Liouville's formula to an arbitrary planar domain. When \(\Omega\) is not simply connected, the authors show that each solution of (1) in an arbitrary domain \(\Omega\subset \mathbb R^2\simeq\mathbb C\) is naturally associated to a Riemannian surface over \(\Omega\) by means of analytic continuation; the representation formula will hold in terms of a meromorphic function defined on a Riemannian surface defined above applying results of \textit{K. S. Chou} and \textit{T. Y. H. Wan} [Pac. J. Math. 163, No. 2, 269--276 (1994; Zbl 0794.35049)]. The authors present an integral formula describing all solutions of Liouville's equation (1) in the punctured disk. Moreover, they also discuss solutions on the plane with one and two punctures.
scientific article; zbMATH DE number 2136806