Cohomogeneity-one Einstein metrics on closed $4$-manifolds

Abstract

We classify the Einstein metrics on closed 4-manifolds whose isometry group has dimension at least 3. Consequently, we show that the cohomogeneity-one Einstein metrics on a closed $4$-manifold are locally symmetric or homothetic to the Page metric on ${\mathbf{CP}}^2♯{\overline{\mathbf{CP}}}^2$.