Torelli-type theorems for gravitational instantons with quadratic volume growth

Abstract

We prove Torelli-type uniqueness theorems for both ${\mathrm{ALG}}^{\ast }$ gravitational instantons and ALG gravitational instantons which are of order 2. That is, the periods uniquely characterize these types of gravitational instantons up to diffeomorphism. We define a period mapping $\mathcal{P}$, which we show is surjective in the ALG cases, and has open image in the ${\mathrm{ALG}}^{\ast }$ cases. We also construct some new degenerations of hyper-Kähler metrics on the K3 surface which exhibit bubbling of ${\mathrm{ALG}}^{\ast }$ gravitational instantons.