Constant Scalar Curvature Metrics on Connected Sums

Journal article

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 $n \geq 3$, which minimizes the total scalar curvature of 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 $\epsilon$ 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.

Authors: Joyce, D

• Publication date: 2003-01-01