Generalized connected sum construction for scalar flat metrics

Abstract

In this paper we will show that the generalized connected sum construction for constant scalar curvature metrics can be extended to the zero scalar curvature case. In particular we want to construct solutions to the Yamabe equation on the generalized connected sum M = M_1 (\sharp_K) M_2 of two compact Riemannian manifolds (M_1,g_1) and (M_2,g_2) with zero constant scalar curvature along a common (isometrically embedded) submanifold (K,g_K) of codimension \geq 3. We present here two kinds of construction. The first one is the basic model and it works for every couple of scalar flat manifolds, but it has a drawback. In fact following this method we are not allowed to choose a scalar flat metric on the generalized connected sum, although the error can be chosen as small as we want. The second construction is an adjustment of the first one which enable us to get a zero scalar curvature metric on the final manifold, but it require the hypothesis that the starting Riemannian manifolds are non Ricci flat.