Gravitational instantons with faster than quadratic curvature decay (I)

TL;DR

This paper classifies gravitational instantons with rapid curvature decay, showing they have known asymptotic types and are biholomorphic to specific complex surfaces, confirming longstanding conjectures in certain cases.

Contribution

It proves that such instantons have known asymptotic types and are biholomorphic to elliptic surfaces in specific cases, confirming Yau's conjecture for ALG and ALH types.

Findings

Instantons must have known ends: ALE, ALF, ALG, or ALH.

In ALG and ALH non-splitting cases, instantons are biholomorphic to elliptic surfaces minus a divisor.

ALF-D_k instantons have an O(4)-multiplet.

Abstract

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-D_k case, it must have an O(4)-multiplet.