It is a well-known result that if is a complete -dimensional Riemannian manifold with , for some , then has to be compact with diameter less than . The proof of this fact can be found in any graduate book about Riemannian geometry and classically relies on the study of Jacobi fields. We propose here an alternative proof of the diameter theorem that relies on the sharp Sobolev inequality proved in the previous Lecture. The beautiful argument goes back to Bakry and Ledoux. We only sketch the main arguments and refer the readers to the original article.
The theorem by Bakry and Ledoux is the following:
Theorem: Assume that for some , we have the inequality,
then is compact with diameter less than .
Combining this with the inequality
that was proved in the previous Lecture gives . When we conclude then by letting and when , we conclude by choosing .
By using a scaling argument it is easy to see that it is enough to prove that if for some ,
The main idea is to apply the Sobolev inequality to the functions which are the extremals functions on the sphere. Such extremals are solutions of the fully non linear PDE
and on the spheres the extremals are explicitly given by
where and is the distance to a fixed point. So, on our manifold , that satisfies the inequality
we consider the functional
where is a function on that satisfies . The first step is to prove a differential inequality on . For , we denote by the differential operator on defined by
Lemma: Denoting , we have
Proof: We denote and , . By the chain-rule and the hypothesis that , we get
From the Sobolev inequality applied to , we thus have,
It is then an easy calculus exercise to deduce our claim
The next idea is then to use a comparison theorem to bound in terms of solutions of the equation
Actually, such solutions are given by
where , and
We have then the following comparison result:
Lemma: Let be such that
and assume that for some . Then for every ,
Using the previous lemma, we see (again we refer to the original article for the details) that implies that and implies that . Iterating this result on the basis of the Sobolev inequality again, we actually have
from which the conclusion easily follows.