The Yamabe problem has been discussed here. Basically, starting from a metric , for a given constant Yamabe wanted to show there always exists a positive function such that the scalar curvature of metric defined to be equals . In terms of PDEs, the scalar curvature satisfies the equation (called Yamabe equation)
where the scalar curvature of metric . It is not hard to see that in the negative and null cases, two solutions of the Yamabe equation with are proportional. Let us prove the following
Theorem. In the negative and null cases, two solutions of the Yamabe equation with are constant .
We first fix a background metric and let be a solution to the Yamabe equation with . That is, the scalar curvature of metric equals .
We now consider a new metric still sitting in the same conformal class of with conformal factor such that its scalar curvature equals . In other words, is also a solution of the Yamabe equation with . Keep in mind
We have two cases.
- Suppose . From the equation above, , that is, is constant.
- Suppose . At a point where is maximum, , thus . Similarly, at a point where is minimum, , thus . Consequently, .
Notice that the uniqueness result in the positive case is no longer true in general. Due to Obata, this is true for Einstein manifolds, that is, when the Ricci curvature and metric are proportional.