Prescribing scalar curvature on Sn

Abstract

on S for n ≥ 3. In the case R is rotationally symmetric, the well-known Kazdan-Warner condition implies that a necessary condition for (1) to have a solution is: R > 0 somewhere and R′(r) changes signs. Then, (a) is this a sufficient condition? (b) If not, what are the necessary and sufficient conditions? These have been open problems for decades. In Chen & Li, 1995, we gave question (a) a negative answer. We showed that a necessary condition for (1) to have a solution is: R′(r) changes signs in the region where R is positive. (2)