On Gromov's theorem and L 2-Hodge decomposition

Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the corresponding L 2-harmonic sections. In particular, some known results concerning Gromov's theorem and the L 2-Hodge decomposition are considerably improved. 1. Introduction. Recall that Hodge's decomposition theorem provides a representation of the de Rham cohomology by the space of harmonic forms over a compact Riemannian manifold. A useful consequence of this theorem is that the pth Betti number b p coincides with the space dimension of harmonic p-forms. This enables one to estimate b p using analytic approaches. A very famous result in the literature is the following Gromov's theorem [15] (see [5] for extensions to Riemannian vector bundles). Throughout the paper, let M be a connected complete Riemannian manifold of dimension d.