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 J. Kazdan and F. 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. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics, 1998.
• Kazdan, J., and Warner F. Scalar curvature and conformal deformation of Riemannian structure. Journal of Differential Geometry. 10 (1975). 113–134.