In this Lecture, we study in further details the connection between volume growth of metric balls, heat kernel upper bounds and the Sobolev inequality. As we shall see, on a manifold with non negative Ricci curvature, all these properties are equivalent one to each other and equivalent to the isoperimetric inequality as well. We start with some preliminaries about geometric measure theory on Riemannian manifolds.
Let be a complete and non compact Riemannian manifold.
In what follows, given an open set we will indicate with the set of vector fields ‘s, on such that .
Given a function we define the total variation of in as
The space endowed with the norm
is a Banach space. It is well-known that is a strict subspace of . It is important to note that when , then , and one has in fact Given a measurable set we say that it has finite perimeter in if . In such case the horizontal perimeter of relative to is by definition
We say that a measurable set is a Caccioppoli set if for any . For instance, if is an open relatively compact set in whose boundary is dimensional sub manifold of , then it is a Caccioppoli set and where is the Riemannian measure on . We will need the following approximation result.
Proposition: Let , then there exists a sequence of functions in such that:
- (i) ;
- (ii) .
If , then the sequence can be taken in .
Our main result of the Lecture is the following result.
Theorem: Let . Let us assume that . then the following assertions are equivalent:
- (1) There exists a constant such that for every , ,
- (2) There exists a constant such that for , ,
- (3) There exists a constant such that for every Caccioppoli set one has
- (4) With the same constant as in (3), for every one has
That (1) (2) follows immediately from the Li-Yau upper Gaussian bound.
The proof that (2) (3) is not straightforward, it relies on the Li-Yau inequality. Let with . By Li-Yau inequality, we obtain
This gives in particular, ,
where we have denoted , . Since , we deduce
By duality, we deduce that for every , ,
Once we have this crucial information we can return to the Li-Yau inequality and infer
Applying this inequality to , with and , if we have
We thus obtain the following basic inequality: for ,
Suppose now that is a bounded Caccioppoli set. But then, , for any bounded open set . It is easy to see that , and therefore . There exists a sequence in satisfying (i) and
(ii) above. Applying the previous inequality to we obtain
Letting in this inequality, we conclude
Observe now that, using , we have
On the other hand,
We thus obtain
We now observe that the assumption (1) implies
Combining these equations we reach the conclusion
Now the absolute minimum of the function , , where , is given by
Applying this observation with , we conclude
The fact that 3) implies 4) is classical geometric measure theory. It relies on the co-area formula that we recall: For every ,
Let now . We have
By using Minkowski inequality, we get then
Finally, we show that . In what follows we let . Let and be such that
Holder inequality, combined with assumption (4), gives for any with compact support
For any and we now let . Clearly such and supp. Since with this choice , the above inequality implies
which, noting that , we can rewrite as follows
where we have let . Notice that .
Iterating the latter inequality we find
From the doubling property for any there exist constants such that with one has
This estimate implies that
Since on the other hand , and , we conclude that
This establishes (1), thus completing the proof