TY - JOUR

T1 - A note on the Kazdan-Warner type condition

AU - Han, Z. C.

AU - Li, Y. Y.

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

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U2 - 10.1016/S0294-1449(16)30105-6

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

M3 - Article

AN - SCOPUS:0030361354

VL - 13

SP - 283

EP - 292

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 3

ER -