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Conformal metrics with prescribed scalar curvature. (English) Zbl 0628.53041
The main results of this paper are as follows:
Theorem 2.3. If \((M^ 3,g)\) is compact with positive scalar curvature, not conformally diffeomorphic to \(S^ 3\), then any function K on M is the scalar curvature of a metric conformal to g.
Theorem 3.3. If \((M^ n,g)\), \(n=3\) or 4, is compact with zero scalar curvature and K is a nonzero function on M, then K is the scalar curvature of a metric conformal to g iff (i) K changes sign and (ii) \(\int\) \(KdV<0.\)
The authors also obtain higher-dimensional versions of these results. The method of proof is analytical.
Reviewer: W.Ballmann

MSC:
53C20 Global Riemannian geometry, including pinching
35J15 Second-order elliptic equations
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