The interplay between the CR ``flatness condition'' and existence results for the prescribed Webster scalar curvature (Q2911042)

On the unit sphere \( \mathbb{S}^{3}\) of \( \mathbb {C}^{2}\) endowed with its standard contact form \( \theta\) and for a given positive \(C^{2}\) function \(K\) on \( \mathbb{S}^{3}\), the authors find conditions which ensure the existence of a contact form \(\tilde{\theta}\) conformal to \(\theta\) having \(K\) as Webster scalar curvature. The condition \textbf{H} that they give is a CR counterpart of a flatness condition for the Riemann settings. After showing the equivalence between the Heisenberg group and \( \mathbb{S}^{3}\) via the Cayley transform and presenting a general framework and several known facts, they prove their result in the final section by using homology and critical points at infinity.