On the isoperimetric quotient over scalar-flat conformal classes

TL;DR

This paper investigates the maximal isoperimetric quotient within scalar-flat conformal classes on compact manifolds, showing it exceeds Euclidean bounds under certain geometric conditions and is attained in those cases.

Contribution

It establishes conditions under which the supremum of the isoperimetric quotient surpasses Euclidean constants and proves its attainability in these scenarios.

Findings

Supremum exceeds Euclidean isoperimetric constant under specified conditions.

Supremum is achieved when conditions (i) or (ii) are met.

Results depend on boundary geometry and Weyl tensor properties.

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.