Toward Best Isoperimetric Constants for $(H^1,BMO)$-Normal Conformal Metrics on ${\mathbb{R}}^n$, $n\ge 3$

Abstract

The aim of this article is: (a) To establish the existence of the best isoperimetric constants for the $(H^1,BMO)$-normal conformal metrics $e^{2u}|dx{|}^2$ on ${\mathbb{R}}^n$, $n\ge 3$, i.e., the conformal metrics with the Q-curvature orientated conditions $$ (-Δ)^{n/2}u\in H^1(\mathbb R^n) & \ u(x)=\hbox{const.}+\frac{\int_{\mathbb R^n}(\log\frac{|\cdot|}{|x-\cdot|})(-Δ)^{n/2} u(\cdot) d\mathcal{H}^n(\cdot)}{2^{n-1}π^{n/2}Γ(n/2)}; $$ (b) To prove that $(n{\omega }_n^{\frac{1}{n}}{)}^{\frac{n}{n-1}}$ is the optimal upper bound of the best isoperimetric constants for the complete $(H^1,BMO)$-normal conformal metrics with nonnegative scalar curvature; (c) To find the optimal upper bound of the best isoperimetric constants via the quotients of two power integrals of Green's functions for the $n$-Laplacian operators $-div(|\nabla u{|}^{n-2}\nabla u)$.

17 pages