Constant scalar curvature metrics on connected sums. (English) Zbl 1026.53019

Author’s 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 \(n\geq 3\), which minimizes the total scalar curvature on this conformal class.
Let \((M^\prime, g')\) and \((M'', g'')\) be compact Riemannian \(n\)-manifolds. We form their connected sum \(M^\prime\#M''\) by removing small balls of radius \(\epsilon\) from \(M^\prime, M''\) and gluing together \(\mathcal S^{n-1}\) boundaries, and make a metric \(g\) on \(M^\prime\#M''\) by joining together \(g^\prime, g''\) with a partition of unity. In this paper, we use analysis to study metrics with constant scalar curvature on \(M^\prime\#M''\) in the conformal class of \(g\). By the Yamabe problem, we may rescale \(g^\prime\) and \(g''\) to have constant scalar curvature \(1\), \(0\), or \(-1\). Thus, there are nine classes, which we handle separately. We show that the constant scalar curvature metrics either develop small ”necks” separating \(M^\prime\) and \(M''\), or one of \(M^\prime\), \(M''\) 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.
Reviewer: W.Mozgawa (Lublin)


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58E11 Critical metrics
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