Prescribing scalar curvature on Sn
We consider the prescribing scalar curvature equation (1) (Formula Presented) 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: (2) R′(r) changes signs in the region where R is positive. Now is this also a sufficient condition? In this paper, we prove that if R(r) satisfies the 'flatness condition', then (2) is the necessary and sufficient condition for (1) to have a solution. This essentially answers question (b). We also generalized this result to non-symmetric functions R. Here the additional 'flatness condition' is a standard assumption which has been used by many authors to guarantee the existence of a solution. In particular, for n = 3, 'non-degenerate' functions satisfy this condition. Based on Theorem 3 in Chen & Li, 1995, we also show that for some rotationally symmetric R, (1) is solvable while none of the solutions is rotationally symmetric. This is interesting in the studying of symmetry breaking.
Chen, Wenxiong, and Congming Li. "Prescribing scalar curvature on Sn." Pacific Journal of Mathematics 199, no. 1 (2001): 61-78.
Pacific Journal of Mathematics