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
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
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
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: