Isoperimetric inequality on CR-manifolds with nonnegative $Q'$-curvature

TL;DR

This paper investigates the role of $Q'$-curvature in controlling the isoperimetric inequality on CR three-manifolds, establishing a link between curvature conditions and geometric inequalities in CR geometry.

Contribution

It demonstrates that nonnegative $Q'$-curvature influences the isoperimetric inequality on CR manifolds and shows that nonnegative Webster curvature at infinity implies the metric is normal.

Findings

$Q'$-curvature controls the isoperimetric inequality.

Nonnegative Webster curvature at infinity implies the metric is normal.

The behavior is analogous to Riemannian four-manifolds.

Abstract

In this paper, we study contact forms on the three- dimensional Heisenberg manifold with its standard CR structure. We discover that the $Q'$-curvature, introduced by Branson, Fontana and Morpurgo [BFM13] on the CR three-sphere and then generalized to any pseudo-Einstein CR three manifold by Case and Yang [CY95], controls the isoperimetric inequality on such a CR-manifold. To show this, we first prove that the nonnegative Webster curvature at infinity deduces that the metric is normal, which is analogous to the behavior on a Riemannian four-manifold.