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Bifurcation of periodic solutions to the singular Yamabe problem on spheres. (English) Zbl 1348.53044

The paper deals with the “Singular Yamabe Problem”: If \((M,g)\) is a compact Riemannian manifold and \(\Lambda \subset M\) a closed subset, find on \(M \setminus \Lambda\) a complete metric \(g'\) conformal to \(g\) and with constant scalar curvature. A valuable review of the literature on this subject is provided. A particularly interesting case is when the manifold \(M\) is the unit round sphere \(S^m\).
Actually, the paper is aiming at obtaining, by means of bifurcation techniques, many families of new periodic solutions in the case when \(\Lambda = S^1\).
In fact, the authors prove that “there exist uncountably many branches of periodic solutions to the singular Yamabe problem on \(S^m \setminus S^1\), for all \(m \geq 5\), having (constant) scalar curvature arbitrarily close to \((m-4) (m-1)\)”.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching
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