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) $9\le n\le 11$ and $\partial M$ has a nonumbilic point; or (ii) $7\le n\le 9$, $\partial M$ 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.

中文翻译：

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标量平坦共形类上等周比的存在定理

令 $(M,g)$ 是维度为 $n$ 的光滑紧致黎曼流形，边界光滑 $\partial M$，允许标量平坦保角度量。我们证明了标量平坦共形类的等周比的上限值严格大于欧几里德空间中等周不等式的最佳常数，因此可以实现，如果 (i) $9\le n\le 11$并且 $\partial M$ 有一个非脐点；或 (ii) $7\le n\le 9$, $\partial M$ 是脐带并且外尔张量在边界上不会完全消失。这是第二位作者和熊的作品 \cite{Jin-Xiong} 的延续。