In this short note I record a proof of the Pohozaev-Schoen identity (see this paper of Schoen, Prop 1.4). This proof is actually standard and I put it here for my own benefit. If I have time I will give some applications of this identity later.
Theorem 1 (Pohozaev-Schoen identity) Let be a compact Riemannian manifold (possibly with boundary) and is a vector field on , then
where is the scalar curvature, is the traceless Ricci tensor and is the Lie derivative of .
We will calculate using a local orthnormal frame. Let be the traceless Ricci tensor. By the (twice contracted) second Bianchi identity , we have
Therefore by integration by parts,
As is a symmetric tensor, we have
(Here we have used the fact that (Exercise). ) Putting this into the above equation, we have
Corollary 2 If is a conformal vector field (see also here), i.e. for some function on , then
In particular, if is closed.
Remark 1 Theorem 1 can be used to prove the DeLellis-Topping inequality.