The Isoperimetric Inequality and Quasiconformal Maps on Manifolds with Finite Total Q-curvature

In this paper, we obtain the isoperimetric inequality on a conformally flat manifold with finite total Q-curvature. This is a higher dimensional analog of Li and Tam’s result [“Complete surfaces with finite total curvature.” Journal of Differential Geometry 33 (1991): 139–68] on surfaces with finite total Gaussian curvature. The main step in the proof is based on the construction of a quasiconformal map the Jacobian of which is suitably bounded.