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We prove Torelli-type uniqueness theorems for both ALG$^*$ gravitationalinstantons and ALG gravitational instantons which are of order $2$. That is,the periods uniquely characterize these types of gravitational instantons up todiffeomorphism. We define a period mapping $\mathscr{P}$, which we show issurjective in the ALG cases, and open in the ALG$^*$ cases. We also constructsome new degenerations of hyperk\"ahler metrics on the K3 surface which exhibitbubbling of ALG$^*$ gravitational instantons.

Torelli-type theorems for gravitational instantons with quadratic volume growth