The Hodge laplacian on manifolds with boundary

This survey paper is an expanded version of seminars given by the authors at the Institut Fourier. Its main scope is to discuss the fîrst positive eigenvalue ii\tP of the Hodge Laplacian actingon differential /?-forms on amanifoldwith boundary. In section 2 we review the Gallot-Meyer and Chanillo-Trêves estimâtes valid for closed manifolds. In section 3 we give the two gênerai inequalities of [G-S] which will imply some new estimâtes for manifolds with boundary. These are given in section 4. More precisely, we first give a lower bound of fU\fP for manifolds whose boundary have some degree of convexity, and then we show that on convex Euclidean domains the first eigenvalue for the absolute conditions is nondecreasingwith respect to the degree: ii\tP ^ l*i,p-iWe then discuss explicit geometrie bounds from [G-S] and [Gl]. In section 5 we first show that the classicalisoperimetric inequalities which are valid for functions do not extend to forms; then we show that the inequality ii\t p ^ £*i,p-1 does not in gênerai, thus justifying the convexity assumptions in section 4. Finally in section 6 we expose a theorem in [G2] which shows that the Hodge spectrum can be prescribed on Euclidean domains.