Conformal metrics of constant scalar curvature with unbounded volumes

Abstract

For $n\ge 25$, we construct a smooth metric $\tilde{g}$ on the standard $n$-dimensional sphere ${\mathbb{S}}^n$ such that there exists a sequence of smooth metrics $\{{\tilde{g}}_k{\}}_{k\in \mathbb{N}}$ conformal to $\tilde{g}$ where each ${\tilde{g}}_k$ has scalar curvature $R_{{\tilde{g}}_k}\equiv 1$ and their volumes $\text{Vol}({\mathbb{S}}^n,{\tilde{g}}_k)$ tend to infinity as $k$ approaches infinity.

62 pages. Final version, Proc. Lond. Math. Soc. to appear