Conformal and isometric embeddings of gravitational instantons

We construct isometric and conformally isometric embeddings of some gravitational instantons in ${\mathbb{R}}^8$ and ${\mathbb{R}}^7$. In particular we show that the embedding class of the Einstein–Maxwell instanton due to Burns is equal to $3$. For ${\mathbb{C}\mathbb{P}}^2$, Eguchi–Hanson and anti-self-dual Taub-NUT we obtain upper and lower bounds on the embedding class.