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.

The solution in higher dimensions

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.

See also

• Prescribed Ricci curvature problem
• Yamabe problem

References

• 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.