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

Abstract

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 $\mathbb{R}^4/\mathbb{Z}_2$. A similar uniqueness result is proved for Calabi's higher-dimensional analogs of the Eguchi-Hanson space among Ricci-flat K\"ahler ALE orbifolds with group at infinity $\mathbb{Z}_m$.