The problem of prescribed critical functions

Abstract

Let $(M,g)$ be a compact Riemannian manifold on dimension $n\ge 4$ not conformally diffeomorphic to the sphere $S^n$. We prove that a smooth function $f$ on $M$ is a critical function for a metric $\tilde{g}$ conformal to $g$ if and only if there exists $x\in M$ such that $f(x)>0$.