# Prescribing scalar curvature on S^{n}

## Abstract

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.

## Department(s)

Mathematics

## Document Type

Article

## DOI

https://doi.org/10.2140/pjm.2001.199.61

## Publication Date

1-1-2001

## Recommended Citation

Chen, Wenxiong, and Congming Li. "Prescribing scalar curvature on Sn." Pacific Journal of Mathematics 199, no. 1 (2001): 61-78.

## Journal Title

Pacific Journal of Mathematics