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)

MSC:

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