Communications in Analysis and Geometry

Volume 11 (2003)

Number 5

Prescribing a Higher Order Conformal Invariant on Sn

Pages: 837 – 858

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

Author

Simon Brendle

Abstract

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 [22] 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 [24] constructed metrics with prescribed scalar curvature on S3. 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.

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Published 1 January 2003