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

TL;DR

This paper proves that for certain high-dimensional scalar-flat conformal classes on compact manifolds with boundary, the supremum of the isoperimetric ratio exceeds the Euclidean best constant, ensuring the existence of extremal metrics.

Contribution

It establishes existence results for extremal metrics in scalar-flat conformal classes under specific geometric conditions, extending previous work.

Findings

Supremum of isoperimetric ratio exceeds Euclidean constant in certain dimensions.

Existence of extremal scalar-flat conformal metrics is guaranteed under given conditions.

Results depend on boundary geometry and dimension, involving nonumbilic points and Weyl tensor conditions.

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.