Uniqueness among scalar-flat K\"ahler metrics on non-compact toric $4$-manifolds

TL;DR

This paper proves the uniqueness of J-complete scalar-flat K"ahler metrics on strictly unbounded non-compact toric surfaces, showing that the asymptotic behavior uniquely determines such metrics, building on previous constructions.

Contribution

It establishes the uniqueness of scalar-flat K"ahler metrics in this setting and links asymptotic behavior to metric determination, extending prior constructions.

Findings

Two families of scalar-flat K"ahler metrics are the only such metrics on strictly unbounded toric surfaces.

Asymptotic behavior uniquely determines the scalar-flat K"ahler metric.

The result confirms the completeness of the constructed metrics in this setting.

Abstract

Abreu-Sena-Dias have constructed two distinct families of scalar-flat K\"ahler non-compact toric metrics using Donaldson's rephrasing of Joyce's construction in action-angle coordinates. In this paper and using the same set-up, we show that these are the only J-complete scalar-flat K\"ahler metrics on any given strictly unbounded toric surface. We also show that the asymptotic behaviour of such a metric determines it uniquely.