Hermitian, Ricci-flat toric metrics on non-compact surfaces à la Biquard-Gauduchon

Abstract

Biquard-Gauduchon have shown that conformally Kähler, Ricci-flat, ALF toric metrics on the complement of toric divisors are: the Taub-NUT metric with reversed orientation, in the Kerr-Taub-bolt family or in the Chen-Teo family. The same authors have also given a unified construction for the above families relying on an axi-symmetric harmonic function on ${\mathbb{R}}^3$. In this work, we reverse this construction and use methods from a paper of the second named author, "Uniqueness among scalar-flat Kähler metrics on non-compact toric 4-manifolds", to show that all conformally Kähler, Ricci-flat, toric metrics on the complement of toric divisors, under some mild assumptions on the associated moment polytope, are among the families above. In particular all such metrics are ALF.

23 pages, 1 figure