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 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.
Highlights
Let (M, g ) and (M, g ) be compact manifolds of dimension n ≥ 3
The connected sum M = M #M is the result of removing a small ball Bn from each manifold, and joining the resulting manifolds at their common boundary n−1
By the proof of the Yamabe problem, every conformal class on M contains a metric with constant scalar curvature
Summary
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. 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.
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More From: International Journal of Mathematics and Mathematical Sciences
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