Gravitational instantons with faster than quadratic curvature decay (III)

TL;DR

This paper classifies and extends the understanding of ALG and ALH gravitational instantons, providing uniqueness results for ALG instantons and a Torelli-type theorem for ALH instantons, advancing the mathematical theory of these geometric objects.

Contribution

It introduces a classification of ALG and ALH gravitational instantons, extending Hein's construction for ALG and proving a Torelli-type theorem for ALH instantons.

Findings

ALG instantons are uniquely characterized by an extended Hein's construction

A Torelli-type theorem is established for ALH instantons

The paper advances the classification theory of gravitational instantons.

Abstract

This is our third paper in a series on the gravitational instantons. In this paper, we classify ALG and ALH gravitational instantons. In ALG case, we extend Hein's construction slightly and show that it's the only ALG gravitational instanton. In ALH case, we prove a Torelli-type theorem.