# A note on the Kazdan-Warner type condition

### Zheng-chao Han

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 USA### YanYan Li

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 USA

## Abstract

This paper addresses the necessary conditions for a function $K(x)$ on $S_{n}$ to be the scalar curvature function of a metric pointwise conformal to the standard metric on $S_{n}$. 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 $S_{n}$. 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 $2≤n≤4$ 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 $S_{n}$.

### Résumé

Nous construisons certaines fonctions positives sur $S_{n}$, $2≤n≤4$, 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 $S_{n}$.

## Cite this article

Zheng-chao Han, YanYan Li, A note on the Kazdan-Warner type condition. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 3, pp. 283–292

DOI 10.1016/S0294-1449(16)30105-6