| Abstract: |
| Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary. Suppose that (M,g) admits 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 on Euclidean space, and consequently is achieved, if either (i) n>=9 and the boundary has a nonumbilic point; or (ii) n>=7, the boundary is umbilic and the Weyl tensor does not vanish at some boundary point. A crucial ingredient in the proof is the expansion of solutions to the conformal Laplacian equation with blowing up Dirichlet boundary conditions. |
|