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
By the divergence theorem, the left hand side of the above equation is just
Thus . If the equality occurs, one deduces that is positive constant which does not satisfy the PDE. Therefore, as required. Consequently, .
Next we prove that
Indeed, multiplying both sides of the PDE with and integrating over , one gets
Again by the divergence theorem,
since cannot be constant. Consequently, one sees that is necessary. In other words, function need to change sign.