Theorem(Aubin)Let be a compact Riemannian manifold with , where is called theYamabe invariantand defined byThen the infimum of the functional is attained. Namely, the Yamabe problem can be solved.

The original proof of Theorem used the subcritical equations. There is another simple proof by Druet and Hebey using the Brezis and Lieb’s lemma.

*Proof*: After passing to a subsequence, we may also assume that there exists such that weakly in , applying the imbedding theorem, we obtain strongly in , furthermore almost everywhere as . In particular, is nonnegative. It easy form the weakly convergence that

for all , where as . We also have that() by Brezis and Lieb’s lemma

Thanks to the sharp Sobolev inequality of Hebey and Vaugon, there exists such that for any ,

Since , it follows that

Since , and since in , we also have that

Hence,

We assume that and note that

this implies that

Then,

as . and since

We obtain that strongly in as . In particular, is a minimizer for and is a weak nonnegative solution of the Yamabe equation

Regularity argument and the maximum principle then give that is smooth and positive. This prove the Theorem.

The terminology Yamabe invariant has just been generalized by Aubin’s student in his PhD dissertation, 2010 I guess. If I am not wrong, this solved the Hebey–Vaugon conjecture. It is great if you could reprocedure this stuff here.

Thank you for your reminding, i know that paper in Arxiv, but not reading carefully, maybe i will write down here in summer vocation. I’m interested in your blog, which also enlighten me. moreover, you are the first commenter of my blog.