Prescribed scalar curvature problem

In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ƒ on M, construct a Riemannian metric on M whose scalar curvature equals ƒ. Due primarily to the work of Jerry Kazdan and Frank Wilson Warner in the 1970s, this problem is well understood.

If the dimension of M is three or greater, then any smooth function ƒ which takes on a negative value somewhere is the scalar curvature of some Riemannian metric. The assumption that ƒ be negative somewhere is needed in general, since not all manifolds admit metrics which have strictly positive scalar curvature. (For example, the three-dimensional torus is such a manifold.) However, Kazdan and Warner proved that if M does admit some metric with strictly positive scalar curvature, then any smooth function ƒ is the scalar curvature of some Riemannian metric.

• Prescribed Ricci curvature problem
• Yamabe problem

• Aubin, Thierry (1998). Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer Nature. doi:10.1007/978-3-662-13006-3. ISBN 9783540607526.
• Kazdan, Jerry; Warner, Frank Wilson (1975). "Scalar curvature and conformal deformation of Riemannian structure". Journal of Differential Geometry. 10: 113–134. doi:10.4310/jdg/1214432678.

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