Gravitational instantons with faster than quadratic curvature decay (II)

TL;DR

This paper advances the understanding of gravitational instantons by proving improved asymptotic decay rates and classifying ALF-D_k instantons as a specific known metric, contributing to the geometric analysis of hyperk"ahler 4-manifolds.

Contribution

It establishes automatic improvements in asymptotic decay rates and classifies ALF-D_k gravitational instantons as a specific known metric, enhancing the classification theory.

Findings

Asymptotic decay rates can be improved automatically.

ALF-D_k gravitational instantons are uniquely identified as the Cherkis-Hitchin-Ivanov-Kapustin-Lindstr"om-Roček metric.

Provides new insights into the structure of hyperk"ahler 4-manifolds.

Abstract

This is our second paper in a series to study gravitational instantons, i.e. complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instanton must be the Cherkis-Hitchin-Ivanov-Kapustin-Lindstr\"om-Ro\v{c}ek metric.