2024/07/23 by Oliveira, Gonçalo, Sena-Dias, Rosa
#53C25 #53C55 #53D20 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.16843
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 ℝ3. 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.