Existence and Conformal Deformation of Metrics With Prescribed Gaussian and Scalar Curvatures

@article{Kazdan1975ExistenceAC,
  title={Existence and Conformal Deformation of Metrics With Prescribed Gaussian and Scalar Curvatures},
  author={Jerry Kazdan and Frank W. Warner},
  journal={Annals of Mathematics},
  year={1975},
  volume={101},
  pages={317}
}
In previous work [11], [13] we have considered the problem of describing the set of Gaussian curvatures (if dim M = 2) and scalar curvatures (if dim M > 3) on a compact, connected, but not necessarily orientable manifold M (see also [12] for the case of open manifolds). We refer the reader to the introductions of [11] and [13] for relevant background and literature. In this paper we give a unified proof that the obvious sign condition demanded by the Gauss Bonnet Theorem, namely, 
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