In this Lecture we consider a complete and -dimensional Riemannian manifold with non negative Ricci curvature. Our goal is to prove the following fundamental result, which is known as the volume doubling property.
Theorem: There exists a constant such that for every and every one has
Actually by suitably adapting the arguments given in this Lecture, the previous result can be extended to the case of negative Ricci curvature as follows:
Theorem: Assume with . There exist positive constants such that for every and every one has
For simplicity, we show the arguments in the case and let the reader work out the arguments in the case .
This result can be obtained from geometric methods as a consequence of the Bishop-Gromov comparison theorem. The proof we give instead only relies on the previous methods and has the advantage to generalize to a much larger class of operators than Laplace-Beltrami on Riemannian manifolds.
The key heat kernel estimate that leads to the doubling property is the following uniform and scale invariant lower bound on the heat kernel measure of balls.
Theorem: There exist an absolute constant , and , depending only on , such that
Proof: We first recall the following result that was proved in a previous Lecture: Let and . Given , which is bounded and such that is Lipschitz, we have
We choose ,and where will later be optimized. Noting that we presently have
we obtain the inequality
In what follows we consider a bounded function on such that almost everywhere on . For any we consider the function defined by
Notice that Jensen’s inequality gives and so we have We now apply the previous inequality to the function , obtaining
Keeping in mind that , we see that Using this observation in combination with the fact that
and switching notation from to , we infer
The latter inequality finally gives
We now optimize the right-hand side of the inequality with respect to . We notice explicitly that the maximum value of the right-hand side is attained at . We find therefore
We now integrate the inequality between and , obtaining
We infer then
Letting we conclude
At this point we let , and consider the function . Since we clearly have it follows that for every one has This gives the lower bound . To estimate the first term in the right-hand side of the latter inequality, we use the previous estimate which gives To make use of this estimate, we now choose , , obtaining . The conclusion follows then easily
We now turn to the proof of the volume doubling property. We first recall the following basic result which is a straightforward consequence of the Li Yau inequality.
Proposition: Let be the heat kernel on . For every and every one has
We are now in position to prove the doubling.
From the semigroup property and the symmetry of the heat kernel we have for any and Consider now a function such that , on and outside . We thus have
If we take , and , we obtain
At this point we use the crucial previous theorem, which gives for some
Combining the latter inequality with the Harnack inequality, we obtain the following on-diagonal lower bound
Applying the Harnack inequality for every we find
Integration over gives
where we have used . Letting , we obtain from this the on-diagonal upper bound . We finally obtain
where we have used once more the Harnack inequality,
which gives .