Gravitational instantons with quadratic volume growth

Abstract

There are two known classes of gravitational instantons with quadratic volume growth at infinity, known as type $ALG$ and ${ALG}^{\ast }$. Gravitational instantons of type $ALG$ were previously classified by Chen–Chen. In this paper, we prove a classification theorem for ${\mathrm{ALG}}^{\ast }$ gravitational instantons. We determine the topology and prove existence of “uniform” coordinates at infinity for both ALG and ${\mathrm{ALG}}^{\ast }$ gravitational instantons. We also prove a result regarding the relationship between ALG gravitational instantons of order $n$ and those of order 2.