Lower bounds for the index of compact constant mean curvature surfaces in [R.sup.3] and [S.sup.3]

Let M be a compact constant mean curvature surface either in [S.sup.3] or [R.sup.3]. In this paper we prove that the stability index of M is bounded from below by a linear function of the genus. As a by-product we obtain a comparison theorem between the spectrum of the Jacobi operator of M and those of Hodge Laplacian of 1-forms on M. Dedicated to the memory of Professor Manfredo Perdigao do Carmo