Analytic classification of toric Kähler instanton metrics in dimension 4.

We classify all 4-dimensional toric scalar-flat Kähler instanton metrics under a simple closure condition. This is done by reducing M 4 to its 2-dimensional metric polygon (Σ 2 , g Σ) , where a synthetic zero scalar curvature condition allows a purely analytic classification, based on solutions of a pair of boundary-degenerate elliptic functions. The proof uses a Liouville theorem for these boundary-degenerate elliptic equations, and we create a Dirichlet boundary-specification theorem for these equations. [ABSTRACT FROM AUTHOR]