Critical metrics on connected sums of Einstein four-manifolds

We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on CP2 and the product metric on S2×S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific Riemannian functional, which depends on the global geometry of the factors. Furthermore, using certain quotients of S2×S2 as one of the gluing factors, critical metrics on several non-simply-connected manifolds are also obtained.