Bifurcation of periodic solutions to the singular Yamabe problem on spheres

Abstract

We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $S^1$ inside $S^m$, $m\geq 5$, that are conformal to the round (incomplete) metric and "periodic" in the sense of being invariant under a discrete group of conformal transformations. These solutions come from bifurcating branches of constant scalar curvature metrics on compact quotients of $S^m \setminus S^1\cong S^{m-2}\times H^2$.