From vanishing theorems to estimating theorems: the Bochner technique revisited

Introduction Organization of the paper SectionsA -Basic Riemannian geometry B -Examples of vanishing and estimating theorems C -More Riemannian geometry D -Proof of Theorem II E -Bochner's theorem and estimating theorems in a more general setting F -Proofs of the estimating theorems (sketch of) G -From vanishing to estimating theorems : an account H -Possible extensions of the methods I -Final comments (geometric vs. analytic methods) Appendices I -The isoperimetric profile II -Using P. Li's lemma III -Application of D. Meyer's lemma IV -Using the heat equation V -J.Moser's iteration scheme VI -Sobolev embedding constants VII -Symmetrization of elliptic or parabolic inequalities References Introduction.In the 1940s, S. Bochner devised an analytic technique to obtain vanishing theorems for some topological or geometric invariants (e.g.Betti numbers, the dimension of the vector space of Killing vector-fields) on a closed (i.e.compact without boundary) Riemannian manifold, under some curvature assumption.As a matter of fact, the word technique might be misleading.On the one hand it is not so easy to explain the technical details of the proofs in which S. Bochner's ideas are used and this is not our purpose here; on the other hand, the ideas are quite simple.Indeed, the basic idea is to show that some object (a harmonic form in the case of Betti numbers, a Killing vector-field,... ) satisfies an elliptic inequality, provided that some curvature assumption is satisfied.The proofs then reduce to applying the maximum principle or to integrating over the manifold.