Existence and non-existence of cohomogeneity one Einstein metrics

This paper derives sufficient conditions for the existence and non-existence of cohomogeneity one Einstein metrics on double disk bundles of two summands type. These conditions are inequalities that only involve geometric data from the principal orbits. The existence theorem yields five infinite families and ten other sporadic examples of new Einstein metrics. The non-existence theorem yields five examples whose principal orbits and singular orbits admit invariant Einstein metrics.