On some manifolds with positive sigma invariants and their realizing conformal classes

We prove that the metric of the Riemannian product $(\text{mb}{S}^k(r_1)\times \text{mb}{S}^{n-k}(r_2),g_k^n)$, $r_1^2+r_2^2=1$, is a Yamabe metric in its conformal class if, and only if, either $g_k^n$ is Einstein, or the linear isometric embedding of this manifold into the standard $n+1$ dimensional sphere is minimal. We combine this result with Simons’ gap theorem to show that, for $2\le k\le n-2$, the conformal class of the product metric with minimal embedding, which is at the upper end of Simons’ gap theorem, realizes the sigma invariant of $\text{mb}{S}^k\times \text{mb}{S}^{n-k}$, and that this is the only class that achieves such a value. Similarly, we use coherent minimal isometric embeddings of suitably scaled standard Einstein metrics $g$ on $\text{mb}{P}^n(\text{mb}R)$, $\text{mb}{P}^n(\text{mb}C)$, and $\text{mb}{P}^n(\text{mb}H)$ into unit spheres, and determine the sigma invariant of these projective spaces, prove that in each case the conformal class $[g]$ realizes it, and that this realizing class is unique.