Constant scalar curvature metrics on connected sums

Abstract

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 sum M ′ # M ″ by removing small balls of radius ϵ from M ′ , M ″ and gluing together the 𝒮 n − 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.