All toric Hermitian ALE gravitational instantons
Bernardo Araneda, James Lucietti
TL;DR
The work proves a sharp uniqueness result for 4D Ricci-flat ALE manifolds with toric Hermitian non‑Kähler structure: the only smooth example is the Eguchi–Hanson instanton (with orientation opposite to the hyper‑Kähler one). It does so by translating the Einstein equations into a Tod form controlled by an axisymmetric harmonic function $V$ on $oldsymbol{R}^3$, then performing a global rod-structure analysis in Weyl–Papapetrou coordinates to impose boundary and regularity conditions. The combination of ALE fall‑offs, conformal Kähler data, and rod regularity eliminates all but the two-rod configuration, which reproduces Eguchi–Hanson; thus these manifolds are self-dual (indeed hyper‑Kähler). The result provides a toric Hermitian, Kähler-free route to a complete ALE classification, complementing recent Hermitian non‑Kähler ALE insights and clarifying the landscape of Ricci-flat toric instantons. Keywords: Eguchi–Hanson, Tod metric, axisymmetric harmonic function, toric Hermitian, ALE instanton, Weyl–Papapetrou, rod structure, conformal Kähler
Abstract
We prove that the only smooth, Ricci flat, ALE instanton with a toric Hermitian non-Kähler structure is the Eguchi-Hanson instanton. The proof is analogous to the classification of toric Hermitian ALF instantons by Biquard and Gauduchon, although we avoid the use of toric Kähler geometry and instead perform a direct global analysis of the Tod form of the metric in Weyl-Papapetrou coordinates. This supports a conjecture by Gibbons and Bando-Kasue-Nakajima which states that any Ricci flat ALE instanton is self-dual.