View source for Prescribed scalar curvature problem

In [[Riemannian geometry]], a branch of [[Mathematics|mathematics]], the '''prescribed scalar curvature problem''' is as follows: given a [[Closed manifold|closed]], smooth manifold ''M'' and a smooth, real-valued function ''ƒ'' on ''M'', construct a Riemannian metric on ''M'' whose [[Scalar curvature|scalar curvature]] equals ''ƒ''.  Due primarily to the work of [[Biography:Jerry Kazdan|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|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&ndash;134.

[[Category:Riemannian geometry]]
[[Category:Mathematical problems]]
[[Category:Curvature (mathematics)|Scalar curvature]]



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