Isolated singularities of solutions to the Yamabe equation in higher dimensions II

This is the second of two papers on isolated singularities of positive solutions to the Yamabe equation on manifolds with nonflat background metrics. In the first paper, we proved that, in dimensions $3\le n\le 29$, every isolated singularity is either removable or the solution is asymptotic to a radial model singular solution in the flat metric on the Euclidean space with origin as its singular point—a Fowler solution. In the present paper, we show that this dimension range is sharp. More precisely, for each $n\ge 30$, we construct a smooth metric on ${\mathbb{R}\mathbb{P}}^n$ and a positive solution of the corresponding Yamabe equation having an isolated singularity, near which the solution blows up along a sequence at a rate much faster than that of Fowler solutions. Moreover, the singular solution defines a complete conformal metric of constant positive scalar curvature and the metric agrees with the round metric to infinite order at the singular point.