Scalar curvature functions in a conformal class of metrics and conformal transformations

Abstract

This article addresses the problem of prescribing the scalar curvature in a conformal class. (For the standard conformal class on the 2 2 -sphere, this is usually referred to as the Nirenberg problem.) Thanks to the action of the conformal group, integrability conditions due to J. L. Kazdan and F. W. Warner are extended, and shown to be universal. A counterexample to a conjecture by J. L. Kazdan on the role of first spherical harmonics in these integrability conditions on the standard sphere is given. Using the action of the conformal groups, some existence results are also given.