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.