Multiplicity results for the prescribed webster scalar curvature on the three CR sphere under ``flatness condition'' (Q661956)

If one considers the unit sphere \( \mathbb S^{3}\) endowed with the standard contact form \(\theta\) and a \(\mathcal C^{2}\) positive function \(K\) on it, the authors investigate suitable conditions on \(K\) for which there exists a contact form \( \tilde {\theta}\) conformal to \( \theta\) such that \(K\) is its Webster scalar curvature: \( K= R_{\tilde {\theta}}\), \(\tilde {\theta}= u^{2} \theta \) where \(u\) is a real positive function on \( \mathbb S^{3}\). They first show that this problem is equivalent to the solution of a differential equation and discuss why the methods based on compactness results cannot be applied here. They finally obtain a positive result if the curvature satisfies a \(CR\) flatness condition by using techniques related to the theory of critical points at infinity.