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In this paper, we study gravitational instantons (i. e. , complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1. Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2. In 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 ALG and ALH cases. 3. In ALF-Dₖ case, it must have an O (4) -multiplet.
Chen et al. (Fri,) studied this question.