Paper detail

Curvature of scalar-flat Kahler metrics on non-compact symplectic toric 4-manifolds

In this paper, we show that the complete scalar-flat Kahler metrics constructed by Abreu and the author on strictly unbounded toric 4-dimensional orbifolds have finite $L^2$ norm of the full Riemannian tensor. In particular, this answers a question of Donaldon's on the corresponding Generalized Taub-NUT metric on $R^4$. This norm is explicitly determined when the underlying toric manifold is the minimal resolution of a cyclic singularity. In the Ricci-flat case corresponding to gravitational instantons, this recovers a recent result of Atiyah-Lebrun.