Some properties of the Yamabe equation in the null case

Let us consider the Yamabe equation in the null case, that is

where is a compact manifold of dimension without boundary. We assume that is a smooth positive solution.

Since the manifold is compact without the boundary, the most simple result is

by integrating both sides of the equation. Now we prove that

.

Indeed, multiplying both sides of the PDE with and integrating over , one obtains