Let be a complete -dimensional Riemannian manifold and, as usual, denote by its Laplace-Beltrami operator. Throughout the Lecture, we will assume that the Ricci curvature of is bounded from below by with . Our purpose is to prove a first important consequence of the Li-Yau inequality: The parabolic Harnack inequality.

**Theorem:*** Let , . For every and ,
*

**Proof:** We first assume that . Let and let , be an absolutely continuous path such that .

We write the Li-Yau inequality in the form

where , and Let us now consider We compute Now, for every , we have

Choosing and using then the Li-Yau inequality yields

By integrating this inequality from to we get as a result.

We now minimize the quantity over the set of absolutely continuous paths such that . By using reparametrization of paths, it is seen that

with equality achieved for where is a unit geodesic joining and . As a conclusion,

Now, from Cauchy-Schwarz inequality we have

and also

.

This proves the inequality when . We can then extend the result to by considering the approximations , where , , and let and

The following result represents an important consequence of the Harnack inequality.

**Corollary:*** Let be the heat kernel on . For every and every one has
*

**Proof:** Let and be fixed. By the hypoellipticity of , we know that . From the semigroup property we

have and . Since we cannot apply the inequality directly to , we consider , where , , and . From Harnack’s inequality we find

.

Letting , by Beppo Levi’s monotone convergence theorem we obtain

The desired conclusion follows by letting

A nice consequence of the parabolic Harnack inequality for the heat kernel is the following lower bound for the heat kernel:

**Proposition:*** For and ,
*

**Proof:** We just need to use the above Harnack inequality with and let using the asymptotics

Observe that when , the inequality is sharp, since it is actually an equality on the Euclidean space !