Isoperimetric inequality, finite total Q-curvature and quasiconformal map

TL;DR

This paper establishes an isoperimetric inequality for conformally flat manifolds with finite total Q-curvature, extending classical results to higher dimensions using quasiconformal maps.

Contribution

It introduces a higher-dimensional analogue of a classical surface result by constructing a quasiconformal map with bounded Jacobian to prove the inequality.

Findings

Proved isoperimetric inequality for conformally flat manifolds with finite total Q-curvature.

Extended Li and Tam's result from surfaces to higher dimensions.

Constructed a quasiconformal map with bounded Jacobian as a key step.

Abstract

In this paper, we obtain the isoperimetric inequality on conformally flat manifold with finite total $Q$-curvature. This is a higher dimensional analogue of Li and Tam's result \cite{L-T} on surfaces with finite total Gaussian curvature. The main step in the proof is based on the construction of a quasiconformal map whose Jacobian is suitably bounded.