Torelli-type theorems for gravitational instantons with quadratic volume growth

We prove Torelli-type uniqueness theorems for both ALG$^*$ 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 $\mathscr{P}$, which we show is surjective in the ALG cases, and open in the ALG$^*$ cases. We also construct some new degenerations of hyperkähler metrics on the K3 surface which exhibit bubbling of ALG$^*$ gravitational instantons.

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