Lower bounds for the index of compact constant mean curvature surfaces in $\mathbb R^3$ and $\mathbb S^3$

Abstract

Let $M$ be a compact constant mean curvature surface either in $\mathbb S^3$ or $\mathbb R^3$. 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 $M$.