Following the previous note about the work of Trudinger, today we talk about the work of Aubin regarding to the Yamabe problem, that is the following simple PDE

In his elegant paper entitlde “Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire” published in *J. Math. Pures Appl.* in 1976, Aubin proved the existence for almost all manifolds for .

By using the notations used in the note about the work of Yamabe, they are

where and

Aubin proved that

Theorem 1. If satisfiesthen the Yamabe problem is solvable where is the volume of the unit sphere in .

In fact, he proved a stronger result saying that in any case, there holds

and the equality occurs if and only if is conformally equivalent to the sphere with standard metric. Having this result, to solve the Yamabe problem, we have only to exhibit a test function such that .

To prove the above estimate, he made use of his breakthrough on the best constant for the Sobolev embedding, actually, it says that

Theorem 2(Best constants for the Sobolev embedding). The Sobolev embedding theorem holds for any complete manifold of dimension with bounded curvature and positive injectivity radius. Moreover, for any , there exists a constant such that everywith .

When , there holds . A standard argument shows that there is a sequence of smooth positive functions (after normalizing) such that

Consequently,

From the definition of we immediately have as claimed.

Now, to prove Theorem 1, Aubin used variational methods. Assuming has unit volume. Indeed, by normalizing, we can use

When , it is clear that is finite since we can use -norm to control -norm. Then it is a standard routine to obtain existence result for . Once we have this, we can send to obtain a sequence of minimizing solutions, say . To obtain some sort of precompactness property for the sequence we observe by using constant functions that . Moreover, for any . Finally, we can control as follows

In other words, the sequence is bounded in . Then, up to subsequences, we have a weak limit . Since solves the subcritical equation, that is to say

we can pass to the limit as (for a detailed explanation, check my paper). After passing to the limit, solves

Using standard regularity theorems, we can conclude that solves

By normalizing, we can change to any constant we want. However, it is necessary to check whether latex u>0$ or not. The maximum principle simply tells us that either everywhere in or . To rule out the latter, we make use of the estimate and here is the only place we need this. To achieve that goal, thanks to the strongly convergence , it is necessary to bound away from zero. For small , Aubin first wrote as follows

Using the equation, we arrive further at

Consequently,

Now thanks to

we can choose small enough in such a way that

This shows that is bounded from below by a positive constant.

As far as we know, the condition is crucial when solving the Yamabe problem. We note that achieves its maximum value only when is the standard unit sphere with standard metric. Since the stereographic projection is a conformal mapping, is conformal to the standard Euclidean space with standard Euclidean metric. Consequently, any conformally flat manifold is globally conformal to . Regardless to this globally set, either the manifold is *locally conformal flat* or *nonlocally conformal flat*. The latter case under the condition was proved by Aubin.

To conclude the paper, Aubin proved the following

Theorem 3. If () is a compact nonlocally conformally flat Riemannian manifold, then

Therefore, the cases that and that is locally conformally flat are still open. To prove this result, Aubin constructed test functions, i.e. he used the following

Eventually, the Weyl tensor comes up in his asymptotic expansion for the test functions, i.e. the sign for the second term is

provided . When the manifold is not compactly locally conformaly flat, the Weyl tensor never vanishes, see this. This helped him to make calculation possible. However, as Aubin had already pointed out that the Yamabe has not only the critical exponent but also has the scalar function in the linear term.

In the latter notes, we show how Schoen overcame this difficulty using the positive mass theorem.

See also:

- The Yamabe problem: A Story
- The Yamabe problem: The work by Hidehiko Yamabe
- The Yamabe problem: The work by Neil Sidney Trudinger
- The Yamabe problem: The work by Thierry Aubin
- The Yamabe problem: The work by Richard Melvin Schoen

## Leave a Reply