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

Abstract

Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂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 ball, and consequently is achieved, if either (i) 9≤n≤11 and ∂M has a nonumbilic point; or (ii) 7≤n≤9, ∂M is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work [12] by the second named author and Xiong.