A Parametrization of Solutions to the Liouville Equation in a Punctured Disc

Abstract

In this paper, we study parametrization of solutions to the Liouville equation near their isolated singular points. We demonstrate that prescribing asymptotic behaviors forms a new type of problem leading to all such solutions. The harmonic functions determining the asymptotic behaviors play the role of ``free data" as the boundary values in the boundary value problems.