Einstein metrics with torus symmetry on $S^4$ and ${\mathbb{R}}^4$

This note gives direct proofs that, under an effective isometric $T^2$ action, Einstein metrics on smooth homotopy four-spheres are round and complete Ricci-flat metrics on smooth manifolds homeomorphic to ${\mathbb{R}}^4$ are Euclidean or positive one-center Taub–NUT, up to isometry, scaling, and orientation reversal.