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Abstract In a previous paper 7, the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kähler. In this article, we consider general Ricci-flat metrics on these spaces that are close to a given such gravitational instanton with respect to a norm that imposes reasonable fall-off conditions at infinity. We prove that any such Ricci-flat perturbation is necessarily Hermitian and carries a bounded, non-trivial Killing vector field. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must actually be one of the known gravitational instantons classified in 7.
Biquard et al. (Fri,) studied this question.