A note on the Kazdan-Warner type condition

Abstract

This paper addresses the necessary conditions for a function $K(x)$ on ${\mathbf{S}}^n$ to be the scalar curvature function of a metric pointwise conformal to the standard metric on ${\mathbf{S}}^n$. The well known necessary conditions are: $K(x)$ is positive somewhere and satisfies the Kazdan–Warner type condition (see the text below). It has remained an outstanding problem for many years whether the above necessary conditions are also sufficient. Recently W. Chen and C. Li ([ChL]) proved that the above conditions are not sufficient by producing changing sign functions $K$ which satisfy the above conditions, but are not scalar curvature functions of any metric pointwise conformal to the standard metric on ${\mathbf{S}}^n$. In their construction, it is essential that $K$ changes sign. In fact, for $n=$2, it follows from the results of [XY] that for the class of positive nondegenerate axisymmetric functions the Kazdan–Warner type condition is actually necessary and sufficient. This brings up a natural question that whether this is a general fact, or it is only so for axisymmetric functions. In this note we answer the above question for $2\le n\le 4$ by producing a family of positive functions $K$ which satisfy the Kazdan–Warner type condition, but nevertheless are not scalar curvature functions of any metric pointwise conformal to the standard metric on ${\mathbf{S}}^n$.

Résumé

Nous construisons certaines fonctions positives sur ${\mathbf{S}}^n$, $2\le n\le 4$, qui satisfont les conditions de type Kazdan–Warner, mais telles qu’elles ne sont pas les courbures scalaires des métriques conformes á la métrique standard de ${\mathbf{S}}^n$.