It is well known that Kronheimer classified all hyperkähler ALE $4$-manifolds. In particular, any such manifold must be diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$ for a finite group $\Gamma \subset SU(2)$.


Given a manifold $M$ which is diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$, is the ALE metric unique on $M$, up to isometry and rescaling?



Hyperkähler ALE $4$-manifolds
It is proved by Kronheimer ( <https://projecteuclid.org/euclid.jdg/1214443066> ) that the ALE metric is unique up to isometry if you fix the Kähler classes. (It is necessary, else you can give different volumes to the various exceptional curves and you get non-isometric metrics).


