Let be a complete -dimensional Riemannian manifold and denote by its Laplace-Beltrami operator. As usual, we denote by the heat semigroup generated by . Throughout the Lecture, we will assume that the Ricci curvature of is bounded from below by . We recall that this is equivalent to the fact that for every ,
Readers knowing Riemannian geometry know that from Bonnet-Myers theorem, the manifold needs to be compact and we therefore expect the semigroup to converge to equilibrium. However for several lectures, our goal will be to not use the Bonnet-Myers theorem, because eventually we shall provide a proof of this fact using semigroup theory. Thus the results in this Lecture will not use the compactness of .
Lemma: The Riemannian measure is finite, i.e. and for every , the following convergence holds pointwise and in ,
Proof: Let , we have
By means of Cauchy-Schwarz inequality, we
Now it is seen from spectral theorem that in we have a convergence , where belongs to the domain of . Moreover . By ellipticity of we deduce that is a smooth function. Since , we have and therefore is constant.
Let us now assume that . This implies in particular that because no constant besides is in . Using then the previous inequality and letting , we infer
Let us assume , and take for the usual localizing sequence . Letting , we deduce which is clearly absurd. As a consequence .
The invariance of implies then ,
and thus . Finally, using the Cauchy-Schwarz inequality, we find that for , , ,
Thus, we also have
Proposition: The following Poincare inequality is satisfied: For ,
Let . We have by assumption Therefore, by integrating the latter inequality we obtain
But we have
Therefore we obtain
By density, this last inequality is seen to hold for every function . It means that the spectrum of lies in . Since from the previous proof the projection of onto the -eigenspace is given by , we deduce that
which is exactly the inequality we wanted to prove
As observed in the proof, the Poincare inequality
is equivalent to the fact that the spectrum of lies in , or in other words that has a spectral gap of size at least . This is Lichnerowicz estimate. It is sharp, because on the -dimensional sphere it is known that and that the first non zero eigenvalue is exactly equal to .
As a basic consequence of the spectral theorem and of the above spectral gap estimate, we also get the rate convergence to equilibrium in for .
Proposition: Let , then for ,
Exercise: By using the Riesz-Thorin interpolation theorem, show that for , and ,
By using duality, prove a corresponding statement when .
As we have just seen, the convergence in of is connected and actually equivalent to the Poincare inequality.
We now turn to the so-called log-Sobolev inequality which is connected to the convergence in entropy for . This inequality is much stronger (and more useful) than the Poincare inequality. To simplify a little the expressions, we assume in the sequel that (Otherwise, just replace by in the following results).
Proposition: For , ,
Proof: By considering instead of , it is enough to show that if is positive,
We now have
Now, we know that
And, from Cauchy-Schwarz inequality, Therefore,
which is the inequality we claimed
We finally prove the entropic convergence of .
Theorem: Let , . For ,
Proof: Let us assume , otherwise we use the following argument with and consider the functional
which by differentiation gives
Using now the log-Sobolev inequality, we obtain
The Gronwall’s differential inequality implies then: