In this Lecture, we are interested in sharp Sobolev inequalities in positive curvature. Let be a complete and -dimensional Riemannian manifold such that where . We assume . As we already know from Lecture 15 , we have , but as we already stressed we do not want to use Bonnet-Myers theorem, since one of our goals will be to recover it by using heat kernel techniques. Without loss of generality, and to simplify the constants, we assume that . Our goal is to prove the following sharp result:

**Theorem:** *For every and ,
*

Our proof follows an argument due to Bakry. We observe that for , the inequality becomes

which is the Poincare inequality with optimal Lichnerowicz constant. For , we get the log-Sobolev inequality

We prove our Sobolev inequality in several steps.

**Lemma:*** For every , there exists a constant such that for every ,
*

**Proof:** Using Jensen’s inequality, it is enough to prove the result for . We already proved the following Li-Yau inequality: For , , , and ,

As a consequence we have

which yields,

We obtain then

This of course implies the following upper bound on the heat kernel,

Using Varopoulos theorem, we deduce therefore that there is constant such that for every ,

We now use the following inequality which is easy to see:

This yields

.

We can now bound

using the Poincare inequality

We now want to prove that the optimal in the previous inequality satisfies . We assume and consider the functional

Classical non linear variational principles on the functional provide then a positive non trivial solution of the equation

Set where is a constant to be later chosen. By the chain rule for diffusion operators, we get

Multiplying by and integrating yields

Now, integrating by parts,

On the other hand, multiplying

by and integrating with respect to yields

Combining the previous computations gives

Hence, we have

We have from Bochner’s inequality,

Once again, is a parameter that will be later decided. Using the chain, to rewrite the previous inequality, leads after tedious computations to

After integration and integration by parts, we see that

where . Combining the previous inequalities we can eliminate the term . Chosing

we see that the coefficient in front of is zero and we are left with

for some constant which is seen to be non-negative as soon as . We conclude