Communications in Analysis and Geometry
Volume 11 (2003)
Prescribing a Higher Order Conformal Invariant on Sn
Pages: 837 – 858
An important problem in differential geometry is to construct conformal metrics on S2 whose Gauss curvature equals a given positive function f. This problem is equivalent to finding a solution of the equation
-Δ0w = f e2w - 1,
where Δ0 denotes the Laplace operator associated with the standard metric g0 on S2. J. Moser  proved that this equation has a solution if the function f satisfies the condition f(x) = f(-x) for all x ∈ S2. The general case was studied by S.-Y. A. Chang, M. Gursky, and P. Yang [10, 11, 14].
A. Bahri and J. M. Coron [4, 5] and R. Schoen and D. Zhang  constructed metrics with prescribed scalar curvature on S3. J. F. Escobar and R. Schoen  studied the prescribed scalar curvature problem on manifolds which are not necessarily conformally equivalent to the standard sphere.
Our aim is to generalize these results to higher dimensions.