Isoperimetric inequality, $Q$-curvature and $A_p$ weights

TL;DR

This paper extends the control of isoperimetric inequality constants from Gaussian curvature in 2D to Branson Q-curvature in higher dimensions, using $A_p$ weights to establish this relationship.

Contribution

It introduces a method to replace Gaussian curvature with Q-curvature for isoperimetric inequalities in higher dimensions via $A_p$ weights.

Findings

Q-curvature can replace Gaussian curvature in isoperimetric control

Established relationship between $A_p$ weights and Q-curvature

Extended 2D results to higher-dimensional conformal manifolds

Abstract

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 conformal flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson $Q$- curvature. We achieve this by exploring the relationship between $A_p$ weights and integrals of the Q-curvature.