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.