Non-existence of Yamabe minimizers on singular spheres

Abstract

We prove that a minimizer of the Yamabe functional does not exist for a sphere ${\mathbb{S}}^n$ of dimension $n\ge 3$, endowed with a standard edge-cone spherical metric of cone angle greater than or equal to $4\pi$, along a great circle of codimension two. When the cone angle along the singularity is smaller than $2\pi$, the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.