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arXiv · 1709.03644

On the isoperimetric quotient over scalar-flat conformal classes

Tianling Jin·Jingang Xiong
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Abstract

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient 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) $n\ge 12$ and $\partial M$ has a nonumbilic point; or (ii) $n\ge 10$, $\partial M$ is umbilic and the Weyl tensor does not vanish at some boundary point.

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Tianling Jin, Jingang Xiong. 2017-09-12. On the isoperimetric quotient over scalar-flat conformal classes. https://arxiv.org/abs/1709.03644

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