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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 ori
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