Scalar curvature on \(S^ n\) and first spherical harmonics (Q1842134)

Let \((S^ n,g_ 0)\) be the unit sphere of \(\mathbb{R}^{n + 1}\) endowed with its standard metric. On one hand, according to the obstructions of Kazdan-Warner and Bourguignon-Ezin, the functions of the type \(1 + h \circ \phi\), where \(h\) is a first spherical harmonic and where \(\phi\) is a conformal diffeomorphism of \(S^ n\), are not the scalar curvature of a metric conformal to \(g_ 0\). On the other hand, we prove that we can associate to each function \(f\) a first spherical harmonic \(h_ f\) and a conformal diffeomorphism \(\phi\) such that \(f - h \circ \phi\) is the scalar curvature of a metric conformal to \(g_ 0\). When \(n = 3\), if \(f\) is symmetrical at one of its maximum points, there exists \(h_ f\) a first spherical harmonic such that \(f - h_ f\) is the scalar curvature of a metric conformal to \(g_ 0\).