Curvature functions for the sphere in pseudohermitian geometry (Q1176130)

By dealing with the CR Yamabe problem, the author studies in the present paper the problem of prescribing arbitrary pseudohermitian scalar curvature \(R\) in the equation: \(\Delta_ bu+{n^ 2\over 4}u-Ru^ a=0\), \(u>0\), on \(S^{2n+1}\), where \(a>1\) is constant and the sublaplacian operator \(\Delta_ b\) is the real part of Kohn's \(\square_ b\) acting on functions. As a consequence of the main result consisting in an integrability condition obtained here, it follows that there are no positive solutions of the above equation either for \(R=f\) if \(a\geq (n+2)/n\), or for \(R=\hbox{const}+f\) if \(a=(n+2)/n\). The main result is then extended to certain pseudohermitian manifolds.