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# Communications in Analysis and Geometry

## Volume 11 (2003)

### Number 5

### Prescribing a Higher Order Conformal Invariant on S^{n}

Pages: 837 – 858

DOI: http://dx.doi.org/10.4310/CAG.2003.v11.n5.a2

#### Author

#### Abstract

An important problem in differential geometry is to construct conformal metrics on S^{2} whose Gauss curvature equals a given positive function *f*. This problem is equivalent to finding a solution of the equation

-Δ_{0}*w* = *f* e^{2w} - 1,

where Δ_{0} denotes the Laplace operator associated with the standard metric *g*_{0} on *S*^{2}. J. Moser [22] proved that this equation has a solution if the function *f* satisfies the condition *f(x) = f(-x)* for all *x ∈ S*^{2}. 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 [24] constructed metrics with prescribed scalar curvature on *S*^{3}. J. F. Escobar and R. Schoen [18] 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.