Gluing Scalar-Flat Manifolds with Vanishing Mean Curvature on the Boundary

We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold $K$, we produce the generalized connected sum along $K$. On this third manifold we produce a family of scalar-flat metrics with small, constant mean curvature on the boundary which are close to the original metrics in the $C^2$ sense. Under extra geometric conditions on the original manifolds, we can arrange for this family to also have vanishing mean curvature on the boundary.