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Scalar curvature functions in a conformal class of metrics and conformal transformations. (English) Zbl 0622.53023
This paper is concerned with the problem of prescribing scalar curvature functions in a conformal class of metrics. An identity of Kazdan-Warner valid for conformal vector fields of the standard conformal sphere is generalized to a compact Riemannian manifold. The result is that for any conformal vector field X on a compact Riemannian manifold (M,g), the scalar curvature \(s_ g\) satisfies the condition \(\int_{M}(X\cdot s_ g)v_ g=0,\) where \(v_ g\) denotes the volume element of the metric g. Thus the Kazdan-Warner identity is not just a special identity related to the sphere, and is in fact universal. The authors also establish the existence of new forbidden functions, not obstructed by the Kazdan-Warner conditions, on the standard sphere. Kazdan-Warner showed in earlier work that certain functions were forbidden as curvature functions on \(S^ 2\). The new functions provide a counterexample to a conjecture of J. L. Kazdan [Semin. differ. geom., Ann. Math. Stud. 102, 143-157 (1982; Zbl 0491.53038), p. 187].
Reviewer: A.Stone

MSC:
53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
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