Generalized connected sum construction for scalar flat metrics

In this paper we construct constant scalar curvature metrics on the generalized connected sum of two compact Riemannian manifolds (M_1,g_1) and (M_2,g_2) along a common Riemannian submanifold (K,g_K), in the case where the codimension of K is at least 3. We present two constructions: the first one produces a family of “small” (in general nonzero) constant scalar curvature metrics on the generalized connected sum of M_1 and M_2. It yields an extension of Joyce’s result for point-wise connected sums in the spirit of our previous issues for nonzero constant scalar curvature metrics. When the initial manifolds are not Ricci flat, and in particular they belong to the (1+) class in the Kazdan–Warner classification, we refine the first construction in order to produce a family of scalar flat metrics on M. As a consequence we get new solutions to the Einstein constraint equations on the generalized connected sum of two compact time symmetric initial data sets, extending the Isenberg–Mazzeo–Pollack gluing construction.