## 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

### Keywords:

singular Yamabe problem; spheres; periodic solutions
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