Einstein constraints gluing and toroidal cusps

Abstract

Isenberg, Mazzeo, and Pollack construct a connected sum gluing for constant mean curvature initial data sets to the vacuum Einstein Field Equations, obtaining new initial data on the joined manifold. The new data set is arbitrarily close to the original data set outside of the gluing region, controlled by the length of the neck joining the manifolds. This approach is here summarized and then partially adapted in an attempt to create a gluing along toroidal cusps. The relevant differences and obstacles for this new gluing are discussed, restricting to the case of compact 3-manifolds and with a focus on the conformal vector Laplacian on toroidal cusps.