A note on the Kazdan-Warner type condition

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Abstract

This paper addresses the necessary conditions for a function K (x) on Sn to be the scalar curvature function of a metric pointwise conformal to the standard metric on Sn. 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 Sn. 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 Sn.

Original languageEnglish (US)
Pages (from-to)283-292
Number of pages10
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume13
Issue number3
DOIs
StatePublished - 1996

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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