An analytical characterization of Eguchi-Hanson space and its higher-dimensional analogs

Let $(M,g)$ be a complete 4-dimensional Ricci-flat ALE orbifold with finitely many orbifold points and group at infinity ${\mathbb{Z}}_2$. We prove that if the $L^2$ kernel of its Lichnerowicz Laplacian has dimension at most 3, then $(M,g)$ is either the Eguchi-Hanson space or the flat orbifold ${\mathbb{R}}^4\mathrm{/}{\mathbb{Z}}_2$. A similar uniqueness result is proved for Calabi’s higher-dimensional analogs of the Eguchi-Hanson space among Ricci-flat Kähler ALE orbifolds with group at infinity ${\mathbb{Z}}_m$.