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arXiv · 1910.01512

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

Xuezhang Chen·Tianling Jin·Yuping Ruan
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Abstract

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.

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Xuezhang Chen, Tianling Jin, Yuping Ruan. 2019-10-03. An existence theorem on the isoperimetric ratio over scalar-flat conformal classes. https://arxiv.org/abs/1910.01512

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