The isoperimetric inequality and Q-curvature

A well-known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformally flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson's Q  -curvature. We achieve this by exploring the relationship between ApAp weights and integrals of the Q-curvature.