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 la
rger 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.
The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(ngeq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal metric of $g_0$
with scalar curvature $1$ and boundary mean curvature $c$. Combining with Z. C. Han and Y. Y. Lis results, we answer this conjecture affirmatively except for the case that $ngeq 8$, the boundary is umbilic, the Weyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.
