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,

Thus

since cannot be constant. Consequently, one sees that is necessary. In other words, function need to change sign.

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