Some applications of scalar curvature deformation in general relativity

The past several years have seen much activity in constructing solutions of the constraint equations by using geometric gluing techniques. These results require an understanding of the scalar curvature operator (and more generally the constraint operator), from the conformal as well as the underdetermined-elliptic points of view. We discuss several applications of these techniques, including the existence of asymptotically simple vacuum spacetimes, and a construction of multi-horizon initial data with trivial topology.