Let be a complete -dimensional Riemannian manifold and, as usual, denote by its Laplace-Beltrami operator. Throughout the Lecture, we will assume again that the Ricci curvature of is bounded from below by . The Lecture is devoted to the proof of a beautiful inequality due to P. Li and S.T. Yau.

Henceforth, we will indicate .

**Lemma:*** Let , and , and consider the function
which is defined on . We have
*

**Proof:** Let for simplicity . A simple computation gives

On the other hand,

Combining these equations we obtain

From the above equation we see that

Observing that

we conclude that

We now turn to an important variational inequality that shall extensively be used throughout these lectures. Given a function and , we let .

Suppose that , and be given. For a function with we define for ,

**Theorem:*** Let and . Given , with , we have
*

**Proof:**Let , . Consider the function

Applying the previous lemma and the curvature-dimension inequality, we obtain

But, we have

and

Therefore we obtain,

We then easily reach the conclusion by using the parabolic comparison theorem in

As a first application the previous result, we derive a family of Li-Yau type inequalities. We choose the function in a such a way that

That is

Integrating the inequality from to , and denoting , we obtain the following result.

**Proposition:*** Let be a smooth function such that We have
*

A first family of interesting inequalities may be obtained with the choice In this case we have and . In particular, we therefore proved the celebrated Li-Yau inequality:

**Theorem:*** If , . For and , we have
*

In the case, and , it reduces to the beautiful sharp inequality:

.

Although in the sequel, we shall first focus on the case , let us presently briefly discuss the case .

Using the Li-Yau inequality with leads to the Bakry-Qian inequality:

Also, by using

we obtain the following inequality that shall be later used in the lectures: