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

Abstract

ABSTRACT. This article addresses the problem of prescribing the scalar curvature in a conformal class. (For the standard conformal class on the 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.O. Statements of results. Recently, the study of functions which, on a connected Coo n-manifold M without boundary, are scalar curvatures of complete Riemannian metrics has drawn special attention among both analysts and geome- Lawson (cf. [6]). Nevertheless, one among the oldest questions related to scalar curvature functions remains unsolved, the Nirenberg problem, namely "describe all curvature functions on the 2-sphere conformal to a standard one." In this article, we shall address this problem. Since it concerns the most familiar compact manifold 8 2 with its standard conformal class, it was expected to be a simple case of the more involved problem: "On a complete n-dimensional manifold, describe the scalar curvature functions in a given conformal class." (For recent developments on the subject, see [2, 4 and 5] [8]), J. L. Kazdan and F. Warner showed that, even if they were positive somewhere (as made necessary by the Gauss-Bonnet theorem), monotonic functions of the distance to a point were forbidden as curvature functions. This followed from an identity satisfied by first spherical harmonics on 8 2 , which, when correctly stated, generalized to the n-sphere 8 n . More recently, T. Aubin pushed further sharp Sobolev estimates for constrained functions by considering certain submanifolds of the Sobolev space H 1 (M). In this way, in [1], he

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