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 $\mathbb{CP}^2$ and the product metric on $S^2 \times S^2$. Using these metrics in various gluing configurations, 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 $S^2 \times S^2$ as one of the gluing factors, critical metrics on several non-simply-connected manifolds are also obtained.