Constant scalar curvature metrics on connected sums

The Yamabe problem (proved in 1984) guarantees the existence of a metric of constant scalar curvature in each conformal class of Riemannian metrics on a compact manifold of dimension n≥3, which minimizes the total scalar curvature on this conformal class. Let (M′,g′) and (M″,g″) be compact Riemannian n-manifolds. We form their connected sumM′#M″ by removing small balls of radius ϵ from M′, M″ and gluing together the 𝒮n−1 boundaries, and make a metric g on M′#M″ by joining together g′, g″ with a partition of unity. In this paper, we use analysis to study metrics with constant scalar curvature on M′#M″ in the conformal class of g. By the Yamabe problem, we may rescale g′ and g″ to have constant scalar curvature 1,0, or −1. Thus, there are 9 cases, which we handle separately. We show that the constant scalar curvature metrics either develop small “necks” separating M′ and M″, or one of M′, M″ is crushed small by the conformal factor. When both sides have positive scalar curvature, we find three metrics with scalar curvature 1 in the same conformal class.