Acta Mathematica

In this paper, we study gravitational instantons (i.e., complete hyperkähler $4$‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑$D_k$ case, it must have an $O(4)$‑multiplet.