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

TL;DR

This paper proves that certain scalar-flat Kahler metrics on non-compact toric 4-manifolds have finite Riemannian tensor norm, providing explicit calculations and answering a question related to gravitational instantons.

Contribution

It demonstrates the finiteness of the Riemannian tensor norm for scalar-flat Kahler metrics on specific non-compact toric 4-manifolds and explicitly computes this norm in key cases.

Findings

Finite $L^2$ norm of the Riemannian tensor for these metrics.

Explicit norm calculations for minimal resolutions of cyclic singularities.

Connection to gravitational instantons and previous results by Atiyah-Lebrun.

Abstract

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.