Isoperimetric inequality on CR-manifolds with nonnegative Q⁰-curvature

In this paper we study contact forms on the three-dimensional Heisenberg manifold with its standard CR structure. We discover that the Q⁰-curvature, introduced by Branson, Fontana and Morpurgo [3] on the CR three-sphere and then generalized to any pseudo-Einstein CR three-manifold by Case and Yang [6], controls the isoperimetric inequality on such a CR-manifold. As the first and important step to show this, we prove that the nonnegative Webster curvature at infinity implies that the metric is normal, which is analogous to the behavior on a Riemannian four-manifold.