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

Abstract

We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by J. Escobar in the mid 90's. We begin with two compact Riemannian manifolds with boundary, each scalar-flat, of vanishing boundary mean curvature, and equipped with a common submanifold K. Under suitable geometric conditions, we produce a 1-parameter family of metrics on the generalized connect sum along K, each of vanishing scalar curvature and constant boundary mean curvature. Assuming an extra non-degeneracy hypothesis, we can arrange for these metrics to have vanishing boundary mean curvature. Moreover, these metrics converge to the original metrics away from the gluing site in the C 2 topology.