Prescribing scalar curvature on Sⁿ

Wenxiong Chen and Congming Li

Vol. 199 (2001), No. 1, 61–78

DOI: 10.2140/pjm.2001.199.61

Abstract

We consider the prescribing scalar curvature equation

on Sⁿ 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:

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.

Milestones

Received: 15 September 1998

Published: 1 May 2001