An existence theorem on the isoperimetric ratio over scalar-flat conformal classes

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $9le nle 11$ and $partial M$ has a nonumbilic point; or (ii) $7le nle 9$, $partial M$ is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work cite{Jin-Xiong} by the second named author and Xiong.