Torelli-type theorems for gravitational instantons with quadratic volume growth

We prove Torelli-type uniqueness theorems for both 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 open in the ALG${}^{\ast }$ cases. We also construct some new degenerations of hyperk"ahler metrics on the K3 surface which exhibit bubbling of ALG${}^{\ast }$ gravitational instantons.